Question

Which of the following are identities? Check all that apply.

A. cos(x + y) + cos(x - y) = 2cos x cosy
B. sin(x + y) - sin(x-y) = 2cosx siny
C. cos(x + y) + cos(x - y) = cos²x - sin²y
D. sin(x - 7T) = - sin x

Answer 1

Options A and D are correct identities, involving sum-to-product and even-odd identities respectively. Option B and C are incorrect as they do not follow the established trigonometric identities.

Step-by-step explanation:

The student has asked which of the following are identities:

• A. cos(x + y) + cos(x - y) = 2cos x cosy
• B. sin(x + y) - sin(x-y) = 2cosx siny
• C. cos(x + y) + cos(x - y) = cos²x - sin²y
• D. sin(x - 7T) = - sin x

Using trigonometric identities:

1. The sum-to-product identities, which include sin(a ± β) = sin a cos β ± cos a sin β, and cos(a ± β) = cos a cos β - sin a sin β.
2. The even-odd identities, such as sin(-x) = -sin x and cos(-x) = cos x.

We can evaluate each option:

• A. This is a correct identity: cos(x + y) + cos(x - y) = 2cos x cosy.
• B. This is not correct; it should be sin(x + y) - sin(x-y) = 2sin x cosy.
• C. This is not correct; it should involve sin x and sin y (not squared).
• D. This is a correct identity: sin(x - 2πn) = - sin x, where n is an integer. So if 7T = 2πn, then it's true.

Therefore, only A and D are correct identities.