Question

Rewrite x" align="absmiddle" class="latex-formula">
so that it involves only the first power of cosine
​

Answer 1

All you need is the double angle identity:

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

So we have

$\cos^4x=(\cos^2x)^2=\left(\dfrac{1+\cos2x}2\right)^2=\dfrac{1+2\cos2x+\cos^22x}4$

Apply the identity again to the squared term:

$\cos^4x=\dfrac{1+2\cos2x+\frac{1+\cos4x}2}4=\dfrac{3+4\cos2x+\cos4x}8$