answer：Okay, so I have this problem about a new operation T defined for x and y as T(x, y) = (mx + ny)(x + 2y), where m and n are non-zero constants. There are three parts to the problem. Let me try to tackle each part step by step.**Part 1: Find the values of m and n.**Given that T(1, -1) = 0 and T(0, 2) = 8. I need to use these two conditions to find m and n.First, let's compute T(1, -1):T(1, -1) = (m*1 + n*(-1))(1 + 2*(-1)) = (m - n)(1 - 2) = (m - n)(-1) = -(m - n) = 0.So, -(m - n) = 0 implies that m - n = 0, which means m = n.Next, compute T(0, 2):T(0, 2) = (m*0 + n*2)(0 + 2*2) = (0 + 2n)(0 + 4) = (2n)(4) = 8n.Given that T(0, 2) = 8, so 8n = 8, which means n = 1.Since m = n, m is also 1.So, m = 1 and n = 1. That seems straightforward.**Part 2: Find the range of values for a such that the system of inequalities has exactly 3 integer solutions.**The system is:T(2p, 2 - p) > 4T(4p, 3 - 2p) ≤ aWe need to find the range of a such that this system has exactly 3 integer solutions for p.First, let's substitute m = 1 and n = 1 into the operation T.So, T(x, y) = (1*x + 1*y)(x + 2y) = (x + y)(x + 2y).Let's compute T(2p, 2 - p):T(2p, 2 - p) = (2p + (2 - p))(2p + 2*(2 - p)) = (2p + 2 - p)(2p + 4 - 2p) = (p + 2)(4).So, T(2p, 2 - p) = 4(p + 2).The inequality is 4(p + 2) > 4, which simplifies to p + 2 > 1, so p > -1.Next, compute T(4p, 3 - 2p):T(4p, 3 - 2p) = (4p + (3 - 2p))(4p + 2*(3 - 2p)) = (4p + 3 - 2p)(4p + 6 - 4p) = (2p + 3)(6).So, T(4p, 3 - 2p) = 6(2p + 3).The inequality is 6(2p + 3) ≤ a, which simplifies to 2p + 3 ≤ a/6, so 2p ≤ (a/6) - 3, so p ≤ ((a/6) - 3)/2 = (a - 18)/12.So, the system of inequalities simplifies to:p > -1p ≤ (a - 18)/12We need to find the range of a such that there are exactly 3 integer solutions for p.Let me think about what integer values p can take. Since p > -1, the smallest integer p can be is 0. Then, the next integers are 1, 2, etc.We need exactly 3 integer solutions. So, p can be 0, 1, 2. That's three integers. So, we need p ≤ (a - 18)/12 to include p = 0, 1, 2 but exclude p = 3.So, p = 2 must satisfy p ≤ (a - 18)/12, and p = 3 must not satisfy it.So, 2 ≤ (a - 18)/12 < 3Multiply all parts by 12:24 ≤ a - 18 < 36Add 18:42 ≤ a < 54So, the range of a is 42 ≤ a < 54.Wait, let me double-check. If a = 42, then (a - 18)/12 = (42 - 18)/12 = 24/12 = 2. So p ≤ 2. So p can be 0, 1, 2.If a = 54, then (54 - 18)/12 = 36/12 = 3. So p ≤ 3. But we don't want p = 3 to be included, so a must be less than 54.So, a must be at least 42 and less than 54.**Part 3: Write down the relationship that m and n satisfy directly when T(x, y) = T(y, x) for any rational numbers x and y, given that x² ≠ y².**So, T(x, y) = T(y, x). Let's write both expressions.T(x, y) = (mx + ny)(x + 2y)T(y, x) = (my + nx)(y + 2x)Set them equal:(mx + ny)(x + 2y) = (my + nx)(y + 2x)Let me expand both sides.Left side:(mx + ny)(x + 2y) = mx*x + mx*2y + ny*x + ny*2y = m x² + 2m x y + n x y + 2n y² = m x² + (2m + n) x y + 2n y²Right side:(my + nx)(y + 2x) = my*y + my*2x + nx*y + nx*2x = m y² + 2m x y + n x y + 2n x² = 2n x² + (2m + n) x y + m y²So, set left side equal to right side:m x² + (2m + n) x y + 2n y² = 2n x² + (2m + n) x y + m y²Subtract the right side from both sides:[m x² + (2m + n) x y + 2n y²] - [2n x² + (2m + n) x y + m y²] = 0Simplify:(m - 2n) x² + (2n - m) y² = 0Factor:(m - 2n)(x² - y²) = 0Given that x² ≠ y², so x² - y² ≠ 0, which means the coefficient must be zero:m - 2n = 0 => m = 2nSo, the relationship is m = 2n.Wait, but in part 1, we found m = n = 1. But here, m = 2n. That seems contradictory. Did I make a mistake?Wait, in part 1, we had specific values for m and n, but in part 3, it's a general relationship when T(x, y) = T(y, x) for any x and y with x² ≠ y². So, in part 1, m and n were specific, but in part 3, it's a general condition.So, the relationship is m = 2n. That's the condition that m and n must satisfy for T(x, y) = T(y, x) for any x and y with x² ≠ y².But in part 1, with m = n = 1, that would mean m = 2n would require 1 = 2*1 = 2, which is not true. So, that suggests that in part 1, the condition T(x, y) = T(y, x) is not necessarily satisfied for all x and y, only for specific x and y given in the problem.So, part 3 is a separate condition, not related to part 1's specific values.Therefore, the relationship is m = 2n.Wait, but in part 1, m = n = 1, but m = 2n would require m = 2, which contradicts. So, perhaps in part 1, the condition T(x, y) = T(y, x) is not required, only the specific values given.Yes, that makes sense. So, part 3 is a separate condition, not dependent on part 1's specific values.So, the relationship is m = 2n.Let me summarize:1. m = 1, n = 12. 42 ≤ a < 543. m = 2nI think that's it.

answer：Okay, so I have this trigonometry problem here. It says that sin α cos α equals 1/8, and α is between 5π/4 and 3π/2. I need to find the value of cos α minus sin α, and the options are given as A: -√3/2, B: √3/2, C: -3/4, D: 3/4.Alright, let me start by understanding the problem. I know that sin α cos α is given, and I need to find cos α - sin α. Hmm, maybe I can use some trigonometric identities here. I remember that sin(2α) = 2 sin α cos α, so maybe I can relate that.Given sin α cos α = 1/8, so multiplying both sides by 2, I get 2 sin α cos α = 1/4. That means sin(2α) = 1/4. Hmm, okay. So 2α would be an angle whose sine is 1/4. But I'm not sure if that helps me directly. Maybe I should think about another identity.Wait, I also remember that (cos α - sin α)^2 can be expanded. Let me try that. So, (cos α - sin α)^2 equals cos² α - 2 sin α cos α + sin² α. I know that cos² α + sin² α equals 1, so that simplifies the expression to 1 - 2 sin α cos α. Given that sin α cos α is 1/8, substituting that in, I get 1 - 2*(1/8) which is 1 - 1/4, so that equals 3/4. Therefore, (cos α - sin α)^2 is 3/4. Taking the square root of both sides, cos α - sin α is either √(3)/2 or -√(3)/2.Now, I need to figure out whether it's positive or negative. For that, I should consider the quadrant where α is located. The problem states that 5π/4 < α < 3π/2. Let me visualize the unit circle. 5π/4 is 225 degrees, which is in the third quadrant, and 3π/2 is 270 degrees, which is on the negative y-axis. So α is in the third quadrant.In the third quadrant, both sine and cosine are negative. So, cos α is negative and sin α is negative. Now, let's think about the values. In the third quadrant, as we go from 5π/4 to 3π/2, the cosine becomes more negative (since it's moving towards -1) and the sine becomes less negative (since it's moving towards 0). So, which one is larger in magnitude, cos α or sin α?At 5π/4, both sine and cosine are equal in magnitude, both equal to -√2/2. As we move towards 3π/2, cosine becomes more negative, while sine becomes less negative. So, for angles between 5π/4 and 3π/2, |cos α| is greater than |sin α|. Therefore, cos α is more negative than sin α is negative.So, if I subtract sin α from cos α, which is cos α - sin α, since both are negative, subtracting a negative is like adding. So, cos α - sin α would be cos α + |sin α|. But since |cos α| > |sin α|, this sum would be negative? Wait, no, let me think again.Wait, cos α is negative, and sin α is negative. So, cos α - sin α is equal to (negative) - (negative). Which is like (negative) + (positive). So, the result could be either positive or negative, depending on the magnitudes.But earlier, I thought that |cos α| > |sin α|, so cos α is more negative than sin α is negative. So, if I have cos α - sin α, that is equivalent to (negative number) - (negative number). Let me denote cos α as -a and sin α as -b, where a and b are positive numbers. Then, cos α - sin α becomes (-a) - (-b) = -a + b = b - a.Given that |cos α| > |sin α|, which means a > b. So, b - a would be negative. Therefore, cos α - sin α is negative.Wait, but earlier when I squared it, I got 3/4, so the square root is either positive or negative. But according to this, it should be negative. So, cos α - sin α is -√3/2.But hold on, let me double-check. Maybe I made a mistake in the quadrant analysis. Let me recall: in the third quadrant, both sine and cosine are negative. So, cos α is negative, sin α is negative. So, cos α - sin α is negative minus negative, which is negative plus positive. So, it's cos α + |sin α|.But since |cos α| > |sin α|, cos α is more negative, so adding |sin α| (which is positive) to cos α (which is negative) would result in a negative number? Or is it the other way around?Wait, let me take an example. Suppose cos α is -0.8 and sin α is -0.6. Then, cos α - sin α is -0.8 - (-0.6) = -0.8 + 0.6 = -0.2, which is negative. So, in this case, it's negative.But in another example, if cos α is -0.5 and sin α is -0.5, then cos α - sin α is -0.5 - (-0.5) = 0. So, it's zero.Wait, but in our case, α is between 5π/4 and 3π/2. At 5π/4, both sine and cosine are equal in magnitude, but as we move towards 3π/2, cosine becomes more negative, and sine becomes less negative. So, in the interval, |cos α| is greater than |sin α|, so cos α is more negative than sin α is negative.Therefore, cos α - sin α is cos α + |sin α|, but since |cos α| > |sin α|, the result is still negative. So, cos α - sin α is negative.But wait, earlier when I squared it, I got 3/4, so the square root is either positive or negative. But according to this, it's negative. So, the answer should be -√3/2, which is option A.But hold on, in my initial calculation, I thought that since |cos α| > |sin α|, cos α - sin α is negative. But let me think again.Wait, cos α is negative, sin α is negative. So, cos α - sin α is equal to (negative) - (negative) = negative + positive. So, depending on which is larger in magnitude, it could be positive or negative.But in the third quadrant, as we move from 5π/4 to 3π/2, |cos α| increases from √2/2 to 1, while |sin α| decreases from √2/2 to 0. So, at 5π/4, |cos α| = |sin α|, so cos α - sin α would be zero. But as we move towards 3π/2, |cos α| becomes larger than |sin α|, so cos α - sin α becomes negative.Wait, but at 5π/4, cos α - sin α is zero, but as we move beyond 5π/4 towards 3π/2, cos α - sin α becomes negative. So, in the interval 5π/4 < α < 3π/2, cos α - sin α is negative.But wait, let me test with a specific angle. Let's take α = 3π/2 - ε, where ε is a small positive number approaching zero. So, cos α is approaching -1, and sin α is approaching 0 from the negative side. So, cos α - sin α is approaching -1 - 0 = -1, which is negative.Another test: take α = 5π/4 + ε, where ε is a small positive number. So, cos α is slightly less than -√2/2, and sin α is slightly less than -√2/2. So, cos α - sin α is (-√2/2 - ε) - (-√2/2 - ε) = (-√2/2 - ε) + √2/2 + ε = 0. Wait, that can't be right. Wait, maybe I messed up the signs.Wait, if α is slightly more than 5π/4, then both cos α and sin α are slightly less than -√2/2. So, cos α is -√2/2 - ε, and sin α is -√2/2 - ε. So, cos α - sin α is (-√2/2 - ε) - (-√2/2 - ε) = (-√2/2 - ε) + √2/2 + ε = 0. So, at angles just above 5π/4, cos α - sin α is zero. But as we move towards 3π/2, cos α becomes more negative, and sin α becomes less negative, so cos α - sin α becomes negative.Wait, but in the interval 5π/4 < α < 3π/2, is cos α - sin α always negative? Let me check at α = 3π/2 - π/6 = 4π/3. So, cos(4π/3) = -1/2, sin(4π/3) = -√3/2. So, cos α - sin α = (-1/2) - (-√3/2) = (-1/2) + √3/2 ≈ (-0.5) + 0.866 ≈ 0.366, which is positive. Wait, that's positive. Hmm, that contradicts my earlier conclusion.Wait, so at 4π/3, which is 240 degrees, cos α is -1/2, sin α is -√3/2. So, cos α - sin α is (-1/2) - (-√3/2) = (-1/2) + √3/2 ≈ 0.366, which is positive. So, in this case, cos α - sin α is positive.But earlier, I thought that in the interval 5π/4 < α < 3π/2, cos α - sin α is negative. But at 4π/3, which is in that interval, it's positive. So, my earlier reasoning must be flawed.Wait, let me think again. Maybe I need to consider the exact values and the behavior of the functions.Let me plot the functions cos α and sin α in the third quadrant. At 5π/4, both are equal to -√2/2. As α increases towards 3π/2, cos α decreases towards -1, and sin α increases towards 0. So, cos α becomes more negative, while sin α becomes less negative.So, cos α - sin α is equal to (cos α) - (sin α). Since both are negative, subtracting a negative is adding a positive. So, it's like (negative) + (positive). The result depends on which term has a larger magnitude.At 5π/4, both are equal in magnitude, so cos α - sin α is zero. As α increases beyond 5π/4, cos α becomes more negative, while sin α becomes less negative. So, the magnitude of cos α increases, while the magnitude of sin α decreases.Therefore, after 5π/4, cos α - sin α becomes negative because cos α is more negative than sin α is negative. Wait, but at 4π/3, which is 240 degrees, cos α is -1/2 and sin α is -√3/2. So, cos α - sin α is (-1/2) - (-√3/2) = (-1/2) + √3/2 ≈ 0.366, which is positive. So, that contradicts the idea that it's negative.Wait, maybe I made a mistake in the direction. Let me think about it again. If cos α is more negative than sin α, then cos α - sin α is (more negative) - (less negative) = more negative + positive. So, depending on the magnitudes, it could be positive or negative.Wait, let me take specific values. Suppose α is 225 degrees (5π/4), cos α = sin α = -√2/2 ≈ -0.707. So, cos α - sin α = (-0.707) - (-0.707) = 0.Now, take α = 240 degrees (4π/3). Cos α = -1/2 = -0.5, sin α = -√3/2 ≈ -0.866. So, cos α - sin α = (-0.5) - (-0.866) = (-0.5) + 0.866 ≈ 0.366, which is positive.Wait, so at 240 degrees, cos α - sin α is positive. But earlier, I thought that as we move from 5π/4 to 3π/2, cos α becomes more negative, and sin α becomes less negative, so cos α - sin α becomes negative. But in reality, at 240 degrees, it's positive.Wait, maybe I need to find the point where cos α - sin α changes sign. Let me set cos α - sin α = 0. That would happen when cos α = sin α, which occurs at α = 5π/4 and α = π/4 + 2πk, etc. But in the interval 5π/4 < α < 3π/2, cos α is less than sin α because cos α is more negative. So, cos α - sin α is negative?Wait, no, because when cos α is more negative than sin α, cos α - sin α is (more negative) - (less negative) = more negative + positive. So, it depends on which is larger in magnitude.Wait, let me think in terms of numbers. Suppose cos α = -0.8, sin α = -0.6. Then, cos α - sin α = (-0.8) - (-0.6) = (-0.8) + 0.6 = -0.2, which is negative.But in the case of 4π/3, cos α = -0.5, sin α = -0.866. So, cos α - sin α = (-0.5) - (-0.866) = (-0.5) + 0.866 ≈ 0.366, which is positive.Wait, so in this case, even though cos α is more negative than sin α, the result is positive because the magnitude of sin α is larger. Wait, but in the first example, cos α was more negative, and the result was negative. In the second example, cos α was less negative, and the result was positive.Wait, I'm getting confused. Let me clarify.When cos α is more negative than sin α, meaning |cos α| > |sin α|, then cos α - sin α = (negative) - (negative) = negative + positive. If |cos α| > |sin α|, then the negative term is larger in magnitude, so the result is negative. If |cos α| < |sin α|, then the positive term is larger, so the result is positive.But in the interval 5π/4 < α < 3π/2, |cos α| starts equal to |sin α| at 5π/4, then increases beyond that, while |sin α| decreases. So, after 5π/4, |cos α| becomes greater than |sin α|, meaning cos α - sin α becomes negative.But wait, at 4π/3, which is 240 degrees, |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866. So, |cos α| < |sin α|, meaning cos α - sin α is positive. But 4π/3 is in the interval 5π/4 < α < 3π/2? Wait, 5π/4 is 225 degrees, 4π/3 is 240 degrees, and 3π/2 is 270 degrees. So, yes, 4π/3 is in that interval.Wait, so in that case, |cos α| is less than |sin α|, so cos α - sin α is positive. But earlier, I thought that after 5π/4, |cos α| becomes greater than |sin α|, making cos α - sin α negative. But at 4π/3, it's positive. So, my initial assumption must be wrong.Wait, let me think about the behavior of |cos α| and |sin α| in the interval 5π/4 to 3π/2.At 5π/4 (225 degrees), |cos α| = |sin α| = √2/2 ≈ 0.707.As α increases towards 3π/2 (270 degrees), |cos α| increases from √2/2 to 1, while |sin α| decreases from √2/2 to 0.So, at some point between 5π/4 and 3π/2, |cos α| becomes greater than |sin α|. Let me find the angle where |cos α| = |sin α|.We know that |cos α| = |sin α| occurs at α = 5π/4 and α = π/4 + π = 5π/4, etc. So, in the interval 5π/4 < α < 3π/2, |cos α| starts equal to |sin α| at 5π/4, then |cos α| increases while |sin α| decreases. So, after 5π/4, |cos α| becomes greater than |sin α|.Wait, but at 4π/3 (240 degrees), |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866, so |cos α| < |sin α|. That means that between 5π/4 and some angle before 3π/2, |cos α| is less than |sin α|, and after that angle, |cos α| becomes greater than |sin α|.Wait, that can't be, because as α increases from 5π/4 to 3π/2, |cos α| increases from √2/2 to 1, while |sin α| decreases from √2/2 to 0. So, at some point, |cos α| must cross |sin α|.Let me find the angle where |cos α| = |sin α| in the third quadrant.We know that tan α = |sin α| / |cos α|. So, when |cos α| = |sin α|, tan α = 1. So, α = arctan(1) = π/4. But in the third quadrant, that would be α = 5π/4.Wait, but that's the starting point. So, after 5π/4, |cos α| increases and |sin α| decreases, so |cos α| becomes greater than |sin α| immediately after 5π/4.But wait, at 4π/3, which is 240 degrees, |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866, so |cos α| < |sin α|. That suggests that my earlier assumption is wrong.Wait, maybe I made a mistake in the direction. Let me think about the actual values.At 5π/4, cos α = sin α = -√2/2 ≈ -0.707.As α increases towards 3π/2, cos α becomes more negative, moving from -√2/2 to -1, while sin α becomes less negative, moving from -√2/2 to 0.So, the magnitude of cos α increases from √2/2 to 1, while the magnitude of sin α decreases from √2/2 to 0.Therefore, at some point after 5π/4, |cos α| becomes greater than |sin α|. Let me find that angle.Let me set |cos α| = |sin α|. So, |cos α| = |sin α| implies that tan α = ±1. In the third quadrant, tan α = 1, so α = 5π/4. So, after 5π/4, |cos α| becomes greater than |sin α|.Wait, but at 4π/3, which is 240 degrees, |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866, so |cos α| < |sin α|. That contradicts the idea that after 5π/4, |cos α| becomes greater than |sin α|.Wait, maybe I'm miscalculating something. Let me check the exact values.At 5π/4 (225 degrees): cos α = -√2/2 ≈ -0.707, sin α = -√2/2 ≈ -0.707.At 3π/2 (270 degrees): cos α = -1, sin α = 0.So, as α increases from 5π/4 to 3π/2, cos α decreases from -√2/2 to -1, and sin α increases from -√2/2 to 0.So, the magnitude of cos α increases from √2/2 to 1, while the magnitude of sin α decreases from √2/2 to 0.Therefore, at some point after 5π/4, |cos α| becomes greater than |sin α|. Let me find that angle.Let me set |cos α| = |sin α|, which is equivalent to tan α = ±1. In the third quadrant, tan α = 1 occurs at α = 5π/4. So, after 5π/4, tan α increases beyond 1, meaning that |sin α| / |cos α| increases beyond 1, implying that |sin α| > |cos α|.Wait, that can't be, because tan α = |sin α| / |cos α|. So, if tan α > 1, then |sin α| > |cos α|. So, after 5π/4, tan α increases beyond 1, meaning that |sin α| > |cos α| until tan α reaches infinity at 3π/2.Wait, but that contradicts the earlier idea that |cos α| increases beyond |sin α|. Wait, no, because tan α = |sin α| / |cos α|. So, if tan α increases beyond 1, that means |sin α| is increasing relative to |cos α|, but in reality, |sin α| is decreasing and |cos α| is increasing.Wait, I'm getting confused. Let me think differently.Let me consider the angle where |cos α| = |sin α| in the third quadrant. That's at 5π/4. After that, as α increases, |cos α| increases and |sin α| decreases. So, after 5π/4, |cos α| is greater than |sin α|.But wait, at 4π/3, which is 240 degrees, |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866, so |cos α| < |sin α|. That suggests that my earlier conclusion is wrong.Wait, maybe I need to plot the functions or use calculus to see where |cos α| and |sin α| cross.Alternatively, let me think about the derivative. The rate at which |cos α| increases versus the rate at which |sin α| decreases.But maybe that's overcomplicating. Let me instead consider specific angles.At 5π/4: |cos α| = |sin α| = √2/2 ≈ 0.707.At 3π/2: |cos α| = 1, |sin α| = 0.So, as α increases from 5π/4 to 3π/2, |cos α| increases from 0.707 to 1, and |sin α| decreases from 0.707 to 0.Therefore, at some point between 5π/4 and 3π/2, |cos α| must cross |sin α| from below to above. Wait, no, because at 5π/4, they are equal, and as α increases, |cos α| increases while |sin α| decreases. So, immediately after 5π/4, |cos α| becomes greater than |sin α|.But wait, at 4π/3 (240 degrees), which is 15 degrees after 5π/4 (225 degrees), |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866, so |cos α| < |sin α|. That suggests that my earlier conclusion is wrong.Wait, maybe I made a mistake in the direction of increase. Let me think about the actual values.Wait, 5π/4 is 225 degrees, 4π/3 is 240 degrees, which is 15 degrees more. So, let me compute |cos α| and |sin α| at 225 degrees, 240 degrees, and 270 degrees.At 225 degrees (5π/4):- cos α = -√2/2 ≈ -0.707- sin α = -√2/2 ≈ -0.707- |cos α| = |sin α| = 0.707At 240 degrees (4π/3):- cos α = -1/2 = -0.5- sin α = -√3/2 ≈ -0.866- |cos α| = 0.5- |sin α| ≈ 0.866At 270 degrees (3π/2):- cos α = -1- sin α = 0- |cos α| = 1- |sin α| = 0So, from 225 degrees to 270 degrees, |cos α| increases from 0.707 to 1, while |sin α| decreases from 0.707 to 0.But at 240 degrees, which is 15 degrees after 225 degrees, |cos α| is 0.5 and |sin α| is 0.866, so |cos α| < |sin α|.Wait, that suggests that between 225 degrees and some angle before 270 degrees, |cos α| is less than |sin α|, and after that angle, |cos α| becomes greater than |sin α|.Wait, that can't be, because as α increases from 225 degrees to 270 degrees, |cos α| is increasing and |sin α| is decreasing, so at some point, |cos α| must cross |sin α| from below to above.Wait, let me find the angle where |cos α| = |sin α| in the third quadrant beyond 5π/4.We know that tan α = |sin α| / |cos α|. So, when |cos α| = |sin α|, tan α = 1. But in the third quadrant, tan α = 1 occurs at α = 5π/4. So, after 5π/4, tan α increases beyond 1, meaning that |sin α| / |cos α| > 1, so |sin α| > |cos α|.Wait, but that contradicts the earlier idea that |cos α| increases beyond |sin α|.Wait, no, because tan α = |sin α| / |cos α|. So, if tan α > 1, then |sin α| > |cos α|. So, after 5π/4, as α increases, tan α increases, meaning |sin α| / |cos α| increases, so |sin α| becomes greater than |cos α|.But wait, that can't be, because as α increases, |cos α| is increasing and |sin α| is decreasing. So, how can |sin α| become greater than |cos α|?Wait, perhaps I'm misunderstanding the behavior. Let me think about the actual functions.In the third quadrant, as α increases from 5π/4 to 3π/2, cos α decreases from -√2/2 to -1, so |cos α| increases from √2/2 to 1. Meanwhile, sin α increases from -√2/2 to 0, so |sin α| decreases from √2/2 to 0.Therefore, |cos α| is increasing, and |sin α| is decreasing. So, at 5π/4, they are equal. After that, |cos α| becomes greater than |sin α| because it's increasing while |sin α| is decreasing.But wait, at 4π/3, which is 240 degrees, |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866, so |cos α| < |sin α|. That suggests that my earlier conclusion is wrong.Wait, maybe I need to plot the functions or use calculus to find where |cos α| = |sin α|.Alternatively, let me consider the derivative of |cos α| and |sin α| with respect to α.But maybe that's overcomplicating. Let me instead consider the angle where |cos α| = |sin α| beyond 5π/4.Wait, but in reality, after 5π/4, |cos α| is greater than |sin α| because |cos α| is increasing and |sin α| is decreasing. So, at 4π/3, which is 240 degrees, |cos α| is 0.5 and |sin α| is √3/2 ≈ 0.866. Wait, that can't be, because 0.5 < 0.866, so |cos α| < |sin α|.Wait, that suggests that my earlier conclusion is wrong, and that |cos α| is less than |sin α| in the interval 5π/4 < α < some angle before 3π/2, and then |cos α| becomes greater than |sin α|.Wait, but how can that be? Because |cos α| is increasing and |sin α| is decreasing, so after 5π/4, |cos α| should be greater than |sin α|.Wait, maybe I'm making a mistake in the direction of increase. Let me think about the actual values.At 5π/4 (225 degrees):- cos α = -√2/2 ≈ -0.707- sin α = -√2/2 ≈ -0.707At 4π/3 (240 degrees):- cos α = -1/2 = -0.5- sin α = -√3/2 ≈ -0.866At 3π/2 (270 degrees):- cos α = -1- sin α = 0So, from 225 degrees to 270 degrees, cos α goes from -0.707 to -1, so it's becoming more negative, meaning |cos α| is increasing.Sin α goes from -0.707 to 0, so it's becoming less negative, meaning |sin α| is decreasing.Therefore, at 225 degrees, |cos α| = |sin α| = 0.707.At 240 degrees, |cos α| = 0.5 and |sin α| = 0.866, so |cos α| < |sin α|.At 270 degrees, |cos α| = 1 and |sin α| = 0, so |cos α| > |sin α|.Therefore, there must be an angle between 240 degrees and 270 degrees where |cos α| = |sin α| again.Wait, but in reality, in the third quadrant, tan α = |sin α| / |cos α|. So, when |cos α| = |sin α|, tan α = 1, which occurs at 5π/4. After that, as α increases, tan α increases beyond 1, meaning |sin α| / |cos α| > 1, so |sin α| > |cos α|.But wait, that contradicts the earlier observation that at 270 degrees, |cos α| = 1 and |sin α| = 0, so |cos α| > |sin α|.Wait, perhaps I made a mistake in the direction. Let me think about it again.Wait, tan α = |sin α| / |cos α|. So, if tan α > 1, then |sin α| > |cos α|. If tan α < 1, then |sin α| < |cos α|.At 5π/4, tan α = 1, so |sin α| = |cos α|.As α increases beyond 5π/4 towards 3π/2, tan α increases beyond 1, so |sin α| > |cos α| until tan α approaches infinity at 3π/2.Wait, but at 3π/2, tan α is undefined because cos α = 0.Wait, but at 3π/2, sin α = 0, so tan α = 0 / -1 = 0. Wait, no, tan α = sin α / cos α. At 3π/2, sin α = -1, cos α = 0, so tan α is undefined (approaches negative infinity from the left and positive infinity from the right).Wait, I'm getting confused again. Let me think about the behavior of tan α in the third quadrant.In the third quadrant, both sin α and cos α are negative, so tan α = sin α / cos α is positive. As α increases from 5π/4 to 3π/2, tan α increases from 1 to positive infinity, because cos α approaches 0 from the negative side, making tan α approach positive infinity.Wait, no, because as α approaches 3π/2 from the left, cos α approaches 0 from the negative side, so tan α = sin α / cos α approaches (negative) / (approaching 0 from negative) = positive infinity.Wait, so tan α increases from 1 to positive infinity as α increases from 5π/4 to 3π/2.Therefore, tan α > 1 in this interval, meaning that |sin α| / |cos α| > 1, so |sin α| > |cos α|.But wait, at 3π/2, sin α = -1 and cos α = 0, so |sin α| = 1 and |cos α| = 0, so |sin α| > |cos α|.But earlier, at 4π/3, which is 240 degrees, |sin α| ≈ 0.866 and |cos α| = 0.5, so |sin α| > |cos α|.Wait, so in the entire interval 5π/4 < α < 3π/2, |sin α| > |cos α|, meaning that cos α - sin α = (negative) - (negative) = negative + positive. Since |sin α| > |cos α|, the positive term is larger, so the result is positive.Wait, but earlier, I thought that at 4π/3, cos α - sin α is positive, which matches this conclusion.But wait, at 3π/2, cos α = -1, sin α = 0, so cos α - sin α = -1 - 0 = -1, which is negative. So, at 3π/2, cos α - sin α is negative.Wait, so there must be a point in the interval where cos α - sin α changes sign from positive to negative.Wait, let me find the angle where cos α - sin α = 0.Set cos α - sin α = 0 => cos α = sin α.In the third quadrant, this occurs at α = 5π/4, where both are equal to -√2/2.But beyond that, as α increases, cos α becomes more negative, and sin α becomes less negative. So, cos α - sin α = (more negative) - (less negative) = more negative + positive.If |cos α| > |sin α|, then the result is negative. If |cos α| < |sin α|, then the result is positive.Wait, but earlier, we saw that in the interval 5π/4 < α < 3π/2, tan α > 1, so |sin α| > |cos α|, meaning that cos α - sin α is positive.But at 3π/2, cos α - sin α is -1, which is negative.So, there must be a point in the interval where cos α - sin α changes from positive to negative.Wait, let me solve for α where cos α - sin α = 0.cos α - sin α = 0 => cos α = sin α => tan α = 1.In the third quadrant, tan α = 1 at α = 5π/4.So, after 5π/4, tan α > 1, meaning |sin α| > |cos α|, so cos α - sin α = (negative) - (negative) = negative + positive. Since |sin α| > |cos α|, the positive term is larger, so the result is positive.But at 3π/2, cos α - sin α = -1, which is negative. So, there must be a point where cos α - sin α changes from positive to negative.Wait, perhaps I made a mistake in assuming that tan α > 1 implies |sin α| > |cos α|. Let me think again.Wait, tan α = |sin α| / |cos α|. So, if tan α > 1, then |sin α| > |cos α|. So, in the interval 5π/4 < α < 3π/2, tan α > 1, so |sin α| > |cos α|.Therefore, cos α - sin α = (negative) - (negative) = negative + positive. Since |sin α| > |cos α|, the positive term is larger, so the result is positive.But at 3π/2, cos α - sin α = -1, which is negative. So, how does that happen?Wait, at 3π/2, sin α = -1, cos α = 0. So, cos α - sin α = 0 - (-1) = 1. Wait, no, cos α is -1 at 3π/2, right? Wait, no, cos(3π/2) is 0, and sin(3π/2) is -1.Wait, no, cos(3π/2) is 0, sin(3π/2) is -1. So, cos α - sin α = 0 - (-1) = 1, which is positive. Wait, but earlier I thought it was -1. That was a mistake.Wait, let me correct that. At 3π/2, cos α = 0, sin α = -1. So, cos α - sin α = 0 - (-1) = 1, which is positive.Wait, so at 3π/2, cos α - sin α is 1, which is positive. So, in the entire interval 5π/4 < α < 3π/2, cos α - sin α is positive.But earlier, I thought that at 3π/2, cos α - sin α is -1, which was incorrect.Wait, let me double-check the values.At 3π/2 (270 degrees):- cos α = 0- sin α = -1- So, cos α - sin α = 0 - (-1) = 1.Yes, that's correct. So, at 3π/2, cos α - sin α is 1, which is positive.Therefore, in the interval 5π/4 < α < 3π/2, cos α - sin α is positive because |sin α| > |cos α|, so the positive term dominates.Therefore, going back to the problem, we have:(cos α - sin α)^2 = 1 - 2 sin α cos α = 1 - 2*(1/8) = 1 - 1/4 = 3/4.So, cos α - sin α = sqrt(3)/2 or -sqrt(3)/2.But since in the interval 5π/4 < α < 3π/2, cos α - sin α is positive, the answer is sqrt(3)/2, which is option B.Wait, but earlier, I thought that at 4π/3, cos α - sin α was positive, which matches this conclusion.So, the correct answer is B: sqrt(3)/2.

answer：Alright, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. It says that the sum of the first 3 terms is 34, the sum of the last 3 terms is 146, and the sum of all the terms is 390. I need to find out how many terms are in this sequence.Okay, let's start by recalling what an arithmetic sequence is. It's a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is 'a' and the common difference is 'd', then the sequence looks like: a, a + d, a + 2d, a + 3d, and so on.Now, the problem gives me some sums. The sum of the first 3 terms is 34. Let's write that out:First term: aSecond term: a + dThird term: a + 2dSo, sum of first 3 terms: a + (a + d) + (a + 2d) = 3a + 3d = 34.Got that. So, equation one is 3a + 3d = 34.Next, the sum of the last 3 terms is 146. Hmm, the last three terms would depend on how many terms there are in total. Let's say the total number of terms is 'n'. Then, the last term would be a + (n - 1)d, the second last term would be a + (n - 2)d, and the third last term would be a + (n - 3)d.So, sum of last 3 terms: [a + (n - 3)d] + [a + (n - 2)d] + [a + (n - 1)d] = 3a + (3n - 6)d = 146.Okay, so equation two is 3a + (3n - 6)d = 146.And the sum of all terms is 390. The formula for the sum of an arithmetic sequence is S = n/2 * (2a + (n - 1)d). So, plugging in the values:n/2 * (2a + (n - 1)d) = 390.Alright, so equation three is n/2 * (2a + (n - 1)d) = 390.Now, I have three equations:1. 3a + 3d = 342. 3a + (3n - 6)d = 1463. n/2 * (2a + (n - 1)d) = 390I need to solve these equations to find 'n'. Let's see how to approach this.First, let's simplify equation one. 3a + 3d = 34 can be divided by 3 to get a + d = 34/3 ≈ 11.333. Hmm, that's a bit messy, but maybe it's okay.Equation two is 3a + (3n - 6)d = 146. Let's see if we can express this in terms of equation one.From equation one, we have 3a + 3d = 34. Let's subtract equation one from equation two:[3a + (3n - 6)d] - [3a + 3d] = 146 - 34Simplify:3a + 3n d - 6d - 3a - 3d = 112The 3a cancels out, and we have:3n d - 6d - 3d = 112Combine like terms:3n d - 9d = 112Factor out 3d:3d(n - 3) = 112So, 3d(n - 3) = 112.Let me write that as equation four: 3d(n - 3) = 112.Now, let's see if we can express 'a' from equation one. From equation one: 3a + 3d = 34, so 3a = 34 - 3d, which means a = (34 - 3d)/3.So, a = (34/3) - d.Alright, now let's look at equation three: n/2 * (2a + (n - 1)d) = 390.Let's substitute 'a' from equation one into equation three.First, compute 2a:2a = 2*(34/3 - d) = 68/3 - 2d.So, 2a + (n - 1)d = 68/3 - 2d + (n - 1)d = 68/3 + (n - 3)d.Therefore, equation three becomes:n/2 * [68/3 + (n - 3)d] = 390.Let me write that as equation five: n/2 * [68/3 + (n - 3)d] = 390.Now, from equation four, we have 3d(n - 3) = 112. Let's solve for d:d = 112 / [3(n - 3)].So, d = 112 / [3(n - 3)].Let's substitute this expression for d into equation five.So, equation five becomes:n/2 * [68/3 + (n - 3)*(112 / [3(n - 3)])] = 390.Simplify inside the brackets:(n - 3) cancels out in the second term:68/3 + 112/3 = (68 + 112)/3 = 180/3 = 60.So, equation five simplifies to:n/2 * 60 = 390.Simplify:n * 30 = 390.Therefore, n = 390 / 30 = 13.So, the number of terms is 13.Wait, let me double-check this because it seems straightforward, but I want to make sure I didn't skip any steps or make any mistakes.Starting from equation four: 3d(n - 3) = 112, so d = 112 / [3(n - 3)].Then, in equation five, substituting d:n/2 * [68/3 + (n - 3)*(112 / [3(n - 3)])] = 390.Indeed, (n - 3) cancels out, leaving 68/3 + 112/3 = 180/3 = 60.So, n/2 * 60 = 390 => n * 30 = 390 => n = 13.That seems correct.Just to be thorough, let's find 'a' and 'd' to make sure everything adds up.From equation one: 3a + 3d = 34 => a + d = 34/3 ≈ 11.333.From equation four: d = 112 / [3(n - 3)] = 112 / [3*(13 - 3)] = 112 / 30 ≈ 3.733.So, d ≈ 3.733.Then, a = 34/3 - d ≈ 11.333 - 3.733 ≈ 7.6.Let's check the sum of the first three terms:a + (a + d) + (a + 2d) = 3a + 3d = 34, which matches.Sum of the last three terms:The last term is a + 12d ≈ 7.6 + 12*3.733 ≈ 7.6 + 44.8 ≈ 52.4.Second last term: a + 11d ≈ 7.6 + 40.866 ≈ 48.466.Third last term: a + 10d ≈ 7.6 + 37.33 ≈ 44.93.Sum: 52.4 + 48.466 + 44.93 ≈ 145.8, which is approximately 146. Close enough, considering rounding.Sum of all terms: n/2 * (2a + (n - 1)d) = 13/2 * (15.2 + 12*3.733) ≈ 6.5 * (15.2 + 44.8) ≈ 6.5 * 60 = 390. Perfect.So, everything checks out. The number of terms is indeed 13.

answer：First, I need to understand the equation (3triangle_4 = triangle2_{11}). Here, (triangle) represents an unknown digit between 0 and 9.To solve for (triangle), I'll convert both sides of the equation into base 10.Starting with the left side, (3triangle_4):- The digit 3 is in the (4^1) place, so its value is (3 times 4 = 12).- The digit (triangle) is in the (4^0) place, so its value is (triangle times 1 = triangle).Adding these together, the left side in base 10 is (12 + triangle).Next, converting the right side, (triangle2_{11}):- The digit (triangle) is in the (11^1) place, so its value is (triangle times 11 = 11triangle).- The digit 2 is in the (11^0) place, so its value is (2 times 1 = 2).Adding these together, the right side in base 10 is (11triangle + 2).Now, I'll set up the equation in base 10:[12 + triangle = 11triangle + 2]Solving for (triangle):[12 + triangle = 11triangle + 2][12 - 2 = 11triangle - triangle][10 = 10triangle][triangle = 1]Therefore, the value of the missing digit (triangle) is 1.

answer：Okay, so I have this expression to simplify: cos(x + y) * sin x + sin(x + y) * cos x. Hmm, that looks a bit complicated at first glance, but maybe I can break it down. Let me think about trigonometric identities that might help here.I remember that there are some angle addition formulas in trigonometry. Specifically, the sine and cosine of sums of angles. The sine addition formula is sin(a + b) = sin a cos b + cos a sin b. Wait a minute, that seems similar to what I have here. Let me write that down to compare:sin(a + b) = sin a cos b + cos a sin b.Now, looking back at my expression: cos(x + y) * sin x + sin(x + y) * cos x. If I let a = x and b = y, then the formula becomes sin(x + y) = sin x cos y + cos x sin y. Hmm, that's not exactly the same as my expression, but it's close. Wait, in my expression, the first term is cos(x + y) * sin x, and the second term is sin(x + y) * cos x. So, it's like I have cos(x + y) multiplied by sin x and sin(x + y) multiplied by cos x. Is there a way to relate this to the sine addition formula?Let me think. Maybe if I consider another identity or perhaps manipulate the expression somehow. Alternatively, maybe I can factor something out. Let me try to see if I can factor out sin x or cos x.Looking at the expression again: cos(x + y) * sin x + sin(x + y) * cos x. It seems like both terms have either sin x or cos x multiplied by another trigonometric function of (x + y). Maybe I can factor out sin x and cos x, but that doesn't seem straightforward.Wait, another thought: perhaps I can use the sine addition formula in reverse. Since the sine addition formula is sin(a + b) = sin a cos b + cos a sin b, maybe my expression is similar to that but with (x + y) and x. Let me try substituting a = x + y and b = x into the sine addition formula.So, sin((x + y) + x) = sin(x + y) cos x + cos(x + y) sin x. That simplifies to sin(2x + y) = sin(x + y) cos x + cos(x + y) sin x. But my original expression is cos(x + y) sin x + sin(x + y) cos x, which is exactly the right-hand side of this equation. So, that means my expression simplifies to sin(2x + y).Wait, but that doesn't seem right because the original expression is cos(x + y) sin x + sin(x + y) cos x, which is similar to the sine addition formula but with (x + y) and x. So, maybe I can think of it as sin((x + y) + x) which is sin(2x + y). But let me double-check that.Alternatively, maybe I can think of it as sin(x + (x + y)) which is also sin(2x + y). So, it seems like my expression simplifies to sin(2x + y). But let me verify this with an example to make sure.Let's pick specific values for x and y to test. Let's say x = 0 and y = 0. Then, cos(0 + 0) * sin 0 + sin(0 + 0) * cos 0 = cos(0) * 0 + sin(0) * 1 = 1 * 0 + 0 * 1 = 0. On the other hand, sin(2*0 + 0) = sin(0) = 0. So, that works.Another example: x = π/2 and y = π/2. Then, cos(π/2 + π/2) * sin(π/2) + sin(π/2 + π/2) * cos(π/2) = cos(π) * 1 + sin(π) * 0 = (-1) * 1 + 0 * 0 = -1. On the other hand, sin(2*(π/2) + π/2) = sin(π + π/2) = sin(3π/2) = -1. So, that also works.Wait, but in the original problem, the expression is cos(x + y) sin x + sin(x + y) cos x, which we've now associated with sin(2x + y). But I'm not sure if that's the simplest form. Maybe there's a simpler way to express this.Let me think again about the sine addition formula. If I have sin(a + b) = sin a cos b + cos a sin b, then my expression is similar but with a = x + y and b = x. So, sin((x + y) + x) = sin(2x + y) = sin(x + y) cos x + cos(x + y) sin x. Therefore, my original expression is equal to sin(2x + y).But is there a way to write this without the 2x? Maybe not, because the expression inherently involves both x and y in a way that combines them into 2x + y. Alternatively, perhaps I can think of it as sin(x + (x + y)) which is the same as sin(2x + y).Wait, but maybe I can factor out something else. Let me try to see if I can factor out sin x or cos x. Let's see:cos(x + y) sin x + sin(x + y) cos x = sin x cos(x + y) + cos x sin(x + y).Hmm, that's the same as before. It seems like the expression is already in a form that directly corresponds to the sine addition formula, just with a different arrangement of terms.Alternatively, maybe I can use the cosine addition formula, but that involves cos(a + b) = cos a cos b - sin a sin b, which doesn't seem directly applicable here.Wait, another thought: perhaps I can express cos(x + y) in terms of cos x and sin x. Let me recall that cos(x + y) = cos x cos y - sin x sin y. Similarly, sin(x + y) = sin x cos y + cos x sin y.So, substituting these into the original expression:cos(x + y) sin x + sin(x + y) cos x = [cos x cos y - sin x sin y] sin x + [sin x cos y + cos x sin y] cos x.Let me expand these terms:First term: [cos x cos y - sin x sin y] sin x = cos x cos y sin x - sin^2 x sin y.Second term: [sin x cos y + cos x sin y] cos x = sin x cos y cos x + cos^2 x sin y.Now, combining these:cos x cos y sin x - sin^2 x sin y + sin x cos y cos x + cos^2 x sin y.Let me rearrange the terms:cos x cos y sin x + cos x cos y sin x - sin^2 x sin y + cos^2 x sin y.Wait, that seems like I have two terms with cos x cos y sin x, so combining them:2 cos x cos y sin x - sin^2 x sin y + cos^2 x sin y.Hmm, this seems more complicated than before. Maybe this approach isn't helpful. Let me try a different way.Going back to the original expression: cos(x + y) sin x + sin(x + y) cos x. I think the key here is to recognize that this expression is actually the sine of a sum of angles. Specifically, if I let a = x + y and b = x, then the expression becomes sin(a + b) = sin a cos b + cos a sin b, which matches our expression.Therefore, cos(x + y) sin x + sin(x + y) cos x = sin((x + y) + x) = sin(2x + y).So, the simplified form is sin(2x + y). Let me double-check this with another example to be sure.Let's take x = π/4 and y = π/4. Then, cos(π/4 + π/4) sin(π/4) + sin(π/4 + π/4) cos(π/4) = cos(π/2) * (√2/2) + sin(π/2) * (√2/2) = 0 * (√2/2) + 1 * (√2/2) = √2/2.On the other hand, sin(2*(π/4) + π/4) = sin(π/2 + π/4) = sin(3π/4) = √2/2. So, that matches as well.Therefore, it seems that the expression simplifies to sin(2x + y). I think that's the simplest form unless there's another identity that can further simplify it, but I don't recall one off the top of my head.Wait, another thought: maybe I can express sin(2x + y) in terms of sin(x + (x + y)) which is the same as sin(2x + y). Alternatively, perhaps using double-angle identities, but that might not necessarily simplify it further.Alternatively, maybe I can write it as sin(x + (x + y)) which is the same as sin(2x + y), but that doesn't seem to simplify it any further.So, I think the simplest form of the expression is sin(2x + y). Therefore, the simplified expression is sin(2x + y).But wait, in the initial problem, the expression was cos(x + y) sin x + sin(x + y) cos x. When I applied the sine addition formula, I got sin(2x + y). But I'm wondering if there's a different way to interpret this expression that might lead to a different simplification.Alternatively, perhaps I can factor out sin x and cos x in a different way. Let me try:cos(x + y) sin x + sin(x + y) cos x = sin x cos(x + y) + cos x sin(x + y).Hmm, that's the same as before. It seems like the expression is directly matching the sine addition formula, just with a different arrangement of terms.Wait, another approach: perhaps using the product-to-sum identities. Let me recall that sin A cos B = [sin(A + B) + sin(A - B)] / 2 and cos A sin B = [sin(A + B) - sin(A - B)] / 2.So, applying these to each term:First term: cos(x + y) sin x = [sin(x + (x + y)) + sin(x - (x + y))] / 2 = [sin(2x + y) + sin(-y)] / 2 = [sin(2x + y) - sin y] / 2.Second term: sin(x + y) cos x = [sin((x + y) + x) - sin((x + y) - x)] / 2 = [sin(2x + y) - sin y] / 2.Now, adding these two terms together:[sin(2x + y) - sin y] / 2 + [sin(2x + y) - sin y] / 2 = [2 sin(2x + y) - 2 sin y] / 2 = sin(2x + y) - sin y.Wait, that's different from what I had before. So, according to this, the expression simplifies to sin(2x + y) - sin y. But earlier, I thought it simplified to sin(2x + y). Which one is correct?Let me test with specific values again. Let's take x = 0 and y = 0. Then, the original expression is cos(0) sin 0 + sin(0) cos 0 = 1*0 + 0*1 = 0. According to the first simplification, sin(2*0 + 0) = 0, which matches. According to the second simplification, sin(0) - sin 0 = 0 - 0 = 0, which also matches.Another example: x = π/2, y = π/2. Original expression: cos(π) sin(π/2) + sin(π) cos(π/2) = (-1)*1 + 0*0 = -1. First simplification: sin(2*(π/2) + π/2) = sin(π + π/2) = sin(3π/2) = -1. Second simplification: sin(2*(π/2) + π/2) - sin(π/2) = sin(3π/2) - 1 = -1 - 1 = -2. Wait, that doesn't match. So, the second simplification must be incorrect.Hmm, that's confusing. So, when I used the product-to-sum identities, I ended up with sin(2x + y) - sin y, but that doesn't match the original expression in the second example. Therefore, that approach must be wrong.Wait, let me double-check the product-to-sum identities. The identity for sin A cos B is [sin(A + B) + sin(A - B)] / 2, and for cos A sin B is [sin(A + B) - sin(A - B)] / 2. So, applying these correctly:First term: cos(x + y) sin x = [sin(x + (x + y)) + sin(x - (x + y))] / 2 = [sin(2x + y) + sin(-y)] / 2 = [sin(2x + y) - sin y] / 2.Second term: sin(x + y) cos x = [sin((x + y) + x) - sin((x + y) - x)] / 2 = [sin(2x + y) - sin y] / 2.Adding them together: [sin(2x + y) - sin y] / 2 + [sin(2x + y) - sin y] / 2 = [2 sin(2x + y) - 2 sin y] / 2 = sin(2x + y) - sin y.But in the second example, this gives sin(3π/2) - sin(π/2) = -1 - 1 = -2, whereas the original expression was -1. So, there's a discrepancy. Therefore, my application of the product-to-sum identities must be incorrect, or perhaps I made a mistake in the algebra.Wait, let me re-examine the product-to-sum identities. Maybe I misapplied them. The identity for sin A cos B is indeed [sin(A + B) + sin(A - B)] / 2, and for cos A sin B is [sin(A + B) - sin(A - B)] / 2. So, applying these correctly:First term: cos(x + y) sin x = [sin(x + (x + y)) + sin(x - (x + y))] / 2 = [sin(2x + y) + sin(-y)] / 2 = [sin(2x + y) - sin y] / 2.Second term: sin(x + y) cos x = [sin((x + y) + x) - sin((x + y) - x)] / 2 = [sin(2x + y) - sin y] / 2.Adding these together: [sin(2x + y) - sin y] / 2 + [sin(2x + y) - sin y] / 2 = [2 sin(2x + y) - 2 sin y] / 2 = sin(2x + y) - sin y.But in the second example, this gives sin(3π/2) - sin(π/2) = -1 - 1 = -2, whereas the original expression was -1. So, clearly, this approach is leading to an incorrect result. Therefore, I must have made a mistake in applying the identities.Wait, perhaps I made a mistake in the signs. Let me double-check the product-to-sum identities. The identity for sin A cos B is [sin(A + B) + sin(A - B)] / 2, and for cos A sin B is [sin(A + B) - sin(A - B)] / 2. So, applying these correctly:First term: cos(x + y) sin x = [sin(x + (x + y)) + sin(x - (x + y))] / 2 = [sin(2x + y) + sin(-y)] / 2 = [sin(2x + y) - sin y] / 2.Second term: sin(x + y) cos x = [sin((x + y) + x) - sin((x + y) - x)] / 2 = [sin(2x + y) - sin y] / 2.Adding these together: [sin(2x + y) - sin y] / 2 + [sin(2x + y) - sin y] / 2 = [2 sin(2x + y) - 2 sin y] / 2 = sin(2x + y) - sin y.But in the second example, this gives sin(3π/2) - sin(π/2) = -1 - 1 = -2, whereas the original expression was -1. So, the product-to-sum approach is leading to an inconsistency. Therefore, I must have made a mistake in applying the identities, or perhaps this approach isn't valid here.Wait, another thought: maybe I should have used different identities or perhaps this expression can't be simplified further using product-to-sum identities. Alternatively, perhaps the initial approach using the sine addition formula was correct, and the product-to-sum approach is leading me astray.Given that the initial approach using the sine addition formula gave consistent results with the examples, I think that the correct simplification is sin(2x + y). Therefore, despite the confusion with the product-to-sum identities, I believe the expression simplifies to sin(2x + y).So, to summarize, the expression cos(x + y) sin x + sin(x + y) cos x simplifies to sin(2x + y) using the sine addition formula.

answer：Okay, so I have this problem about a sequence where the sum of the first n terms is Sₙ. I know that a₁ is 2, and there's this equation given: (n - 2)Sₙ₊₁ + 2aₙ₊₁ = nSₙ for all positive integers n. I need to figure out which of the statements A, B, C, D are correct.First, let me write down what I know:- a₁ = 2- Sₙ is the sum of the first n terms, so Sₙ = a₁ + a₂ + ... + aₙ- The equation given is (n - 2)Sₙ₊₁ + 2aₙ₊₁ = nSₙI need to find out which of the options A, B, C, D are correct. Let's go through each option one by one, but maybe first, I should try to find a general formula for Sₙ or aₙ.Starting with the given equation:(n - 2)Sₙ₊₁ + 2aₙ₊₁ = nSₙHmm, I can express aₙ₊₁ in terms of Sₙ₊₁ and Sₙ because aₙ₊₁ = Sₙ₊₁ - Sₙ. Let me substitute that into the equation:(n - 2)Sₙ₊₁ + 2(Sₙ₊₁ - Sₙ) = nSₙLet me expand this:(n - 2)Sₙ₊₁ + 2Sₙ₊₁ - 2Sₙ = nSₙCombine like terms:[(n - 2) + 2]Sₙ₊₁ - 2Sₙ = nSₙSimplify inside the brackets:(n - 2 + 2) = n, so:nSₙ₊₁ - 2Sₙ = nSₙBring the -2Sₙ to the right side:nSₙ₊₁ = nSₙ + 2SₙFactor out Sₙ on the right:nSₙ₊₁ = Sₙ(n + 2)So, divide both sides by n:Sₙ₊₁ = Sₙ(n + 2)/nHmm, that's a recursive formula for Sₙ. Maybe I can write it as:Sₙ₊₁ / (n + 1) = Sₙ / n * (n + 2)/ (n + 1)Wait, let me see:Sₙ₊₁ = Sₙ(n + 2)/nSo, Sₙ₊₁ / (n + 1) = Sₙ(n + 2)/(n(n + 1))Let me denote Tₙ = Sₙ / (n(n + 1)). Then:Tₙ₊₁ = Sₙ₊₁ / ((n + 1)(n + 2)) = [Sₙ(n + 2)/n] / ((n + 1)(n + 2)) = Sₙ / (n(n + 1)) = TₙSo, Tₙ₊₁ = Tₙ, which means Tₙ is a constant sequence. Therefore, Tₙ = T₁ for all n.Compute T₁:T₁ = S₁ / (1*2) = a₁ / 2 = 2 / 2 = 1So, Tₙ = 1 for all n. Therefore, Sₙ = n(n + 1). So, Sₙ = n² + n.That's statement D. So, D is correct.Now, since Sₙ = n² + n, we can find aₙ. For n ≥ 2, aₙ = Sₙ - Sₙ₋₁.Compute aₙ:aₙ = Sₙ - Sₙ₋₁ = [n² + n] - [(n - 1)² + (n - 1)] = n² + n - (n² - 2n + 1 + n - 1) = n² + n - (n² - n) = n² + n - n² + n = 2nSo, aₙ = 2n for n ≥ 2. But what about n = 1? a₁ is given as 2, which is also 2*1, so aₙ = 2n for all n ≥ 1.So, aₙ = 2n for all n.Now, let's check the options:A: a₂ = 4. Since aₙ = 2n, a₂ = 4. So, A is correct.B: {aₙ / n} is a constant sequence. Since aₙ = 2n, aₙ / n = 2, which is constant. So, B is correct.C: a₇ = 15. But a₇ = 2*7 = 14, not 15. So, C is incorrect.D: Sₙ = n² + n. We already found that Sₙ = n² + n, so D is correct.So, the correct options are A, B, D.**Final Answer**The correct options are boxed{A}, boxed{B}, and boxed{D}.