Question

Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.

cos^3 x

A. 3/4 cos x − 1/4 cos 3x − cos 2x
B. 3/4 cos x − 1/4 cos 3x
C. 3/4 cos x + 1/4 cos 3x + cos 2x
D. 3/4 cos x + 1/4 cos 3x

Answer 1

$\cos^3x=\cos^2x\cdot\cos x=\dfrac{1+\cos2x}2\cdot\cos x=\dfrac12\cos x+\dfrac12\cos2x\cos x$

Recall that

$\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$

from which we can find that

$\cos\alpha\cos\beta=\dfrac{\cos(\alpha+\beta)+\cos(\alpha-\beta)}2$

Then

$\cos2x\cos x=\dfrac{\cos3x+\cos x}2$

and we end up with

$\cos^3x=\dfrac34\cos x+\dfrac14\cos3x$