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=\frac{1+\cos(2x)}2`

Then

`3\cos^4x=3(\cos^2x)^2=3\left(\frac{1+\cos(2x)}2\right)^2=\frac34(1+\cos(2x))^2`

Expand the binomial to get

`3\cos^4x=\frac34\left(1+2\cos(2x)+\cos^2(2x)\right)`

Use the identity again to write

`\cos^2(2x)=\frac{1+\cos(4x)}2`

and so

`3\cos^4x=\frac34\left(1+2\cos(2x)+\frac{1+\cos(4x)}2\right)`

`3\cos^4x=\frac38\left(3+4\cos(2x)+\cos(4x)\right)`