Question

Sin(90+a)*sin(270-a)+cos(a-90)*sin(a-180)​

Answer 1

Решение на фотографии

Answer 2

To simplify the given expression, let's use some trigonometric identities. Specifically, we can use the sum-to-product and difference-to-product identities for trigonometric functions.

1. Sum-to-product identity: sin(A) * sin(B) = (1/2) * [cos(A - B) - cos(A + B)]
2. Difference-to-product identity: sin(A) * cos(B) = (1/2) * [sin(A + B) + sin(A - B)]

Let's apply these identities step-by-step to simplify the given expression:

Given expression: Sin(90+a)*sin(270-a) + cos(a-90)*sin(a-180)

1. Replace sin(90 + a) with cos(a) using the identity sin(90 + A) = cos(A): cos(a) * sin(270 - a) + cos(a - 90) * sin(a - 180)

2. Replace sin(270 - a) with -sin(a) using the identity sin(270 - A) = -sin(A): cos(a) * (-sin(a)) + cos(a - 90) * sin(a - 180)

3. Replace cos(a - 90) with -sin(a) using the identity cos(A - 90) = -sin(A): cos(a) * (-sin(a)) + (-sin(a)) * sin(a - 180)

4. Now, use the difference-to-product identity sin(A) * cos(B) = (1/2) * [sin(A + B) + sin(A - B)]: (1/2) * [sin(a + (a - 180)) + sin(a - (a - 180))]

5. Simplify the arguments inside the sine functions: (1/2) * [sin(2a - 180) + sin(180)]

6. Since sin(180) = 0, the expression becomes: (1/2) * [sin(2a - 180) + 0]

7. Finally, we have: (1/2) * sin(2a - 180)

So, the simplified expression is (1/2) * sin(2a - 180).