Rewrite {cos}^(4) x so that it involves only the first power of cosine ​

Question

Rewrite

`{cos}^(4) x`
so that it involves only the first power of cosine
​

Answer 1

All you need is the double angle identity:

`\cos^2x=\frac{1+\cos2x}2`

So we have

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

Apply the identity again to the squared term:

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