Question

use the power-reducing formulas as many times as possible to rewrite the expression in terms of the first power of the cosine. sin^4 3x cos^2 3x

Answer 1

`\sin^43x\cos^23x`

Pythagorean identity:


`(1-\cos^23x)^2\cos^23x`

Expand:


`(1-2\cos^23x+\cos^43x)\cos^23x`

Distribute:


`\cos^23x-2\cos^43x+\cos^63x`

Half-angle identity for cosine:


`\frac{1+\cos6x}2-2\left(\frac{1+\cos6x}2\right)^2+\left(\frac{1+\cos6x}2\right)^3`

Expand:


`\frac12+\frac12\cos6x-\frac12\left(1+2\cos6x+\cos^26x\right)+\frac18\left(1+3\cos6x+3\cos^26x+\cos^36x\right)`

Simplify:


`\frac18-\frac18\cos6x-\frac18\cos^26x+\frac18\cos^36x`

Half-angle again:


`\frac18-\frac18\cos6x-\frac18\left(\frac{1+\cos12x}2\right)+\frac18\cos6x\left(\frac{1+\cos12x}2\right)`

Simplify:


`\frac1{16}-\frac1{16}\cos6x-\frac1{16}\cos12x+\frac1{16}\cos6x\cos12x`

Angle sum identity for cosine:


`\frac1{16}-\frac1{16}\cos6x-\frac1{16}\cos12x+\frac1{16}\left(\frac{\cos6x+\cos18x}2\right)`

Simplify:


`\frac1{16}-\frac1{32}\cos6x-\frac1{16}\cos12x+\frac1{32}\cos18x`