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"Find an equivelent expression for sin^4(X) in terms of function values of...

Attached is the problem. The example problem that I get is rewriting cos(4X), which coincidentally was what I got for the actual assignment. I'm reviewing it now.

"Find an equivelent expression for sin^4(X) in terms of function values of the sine or cosine of x, 2x, or 4x raised to the first power."

The answer is 3/8 - (cos(2x))/2 + (cos(4x))/8

(I've attached an image where it's written clearer.)

I haven't the first clue what the first step to get to this answer is. My book is no help at all.