Rewrite in terms of the first power of the cosine: sin^2 x cos^2 2x.?

Question

a. (1/4)(1 - cos 2x + cos 4x - cos 2x cos 4x)
b. (1/4)(1 - cos 2x + cos 4x - cos 8x)
c. (1/4)(1 + cos 2x - cos 8x)
d. (1/4)(1 + cos 2x - cos 4x - cos 2x cos 4x)
e. none of these

Answer 1

a. (1/4)(1 - cos 2x + cos 4x - cos 2x cos 4x)

This is obtained as follows, by using the "cos 2z" formula in two different forms:

cos 2z = 2 cos^2 z - 1 = 1 - 2 sin^2 z. Therefore:

sin^2 x = (1 - cos 2x) / 2, and cos^2 2x = (1 + cos 4x) / 2.

Thus sin^2 x cos^2 2x = (1/4) (1 - cos 2x) (1 + cos 4x)

= (1/4)(1 - cos 2x + cos 4x - cos 2x cos 4x)

QED

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