Question

Use the power reducing formula to rewrite the expression in therms of first powers of hte cosines of ultiple angles 3cos^4x.

Answer 1

The particular identity you want to use is

$\cos^2x=\dfrac{1+\cos(2x)}2$

Then

$3\cos^4x=3(\cos^2x)^2=3\left(\dfrac{1+\cos(2x)}2\right)^2=\dfrac34(1+\cos(2x))^2$

Expand the binomial to get

$3\cos^4x=\dfrac34\left(1+2\cos(2x)+\cos^2(2x)\right)$

Use the identity again to write

$\cos^2(2x)=\dfrac{1+\cos(4x)}2$

and so

$3\cos^4x=\dfrac34\left(1+2\cos(2x)+\dfrac{1+\cos(4x)}2\right)$

$3\cos^4x=\dfrac38\left(3+4\cos(2x)+\cos(4x)\right)$