Question

How do you verify `8sin²xcos²x = 1-cos^4(x)`?

Answer 1

This can not be proven because it is not in general true.

As a counter example, consider `x=pi/4`

`color(white)("XXXX")` `sin(x) = 1/sqrt(2)`
`color(white)("XXXX")` `color(white)("XXXX")` `sin^2(x) = 1/2`
`color(white)("XXXX")` `cos(x) = 1/sqrt(2)`
`color(white)("XXXX")` `color(white)("XXXX")` `cos^2(x) = 1/2`
`color(white)("XXXX")` `color(white)("XXXX")` `cos^4(x) = 1/4`

`8 sin^2xcos^2x = 8 * 1/2*1/2 = 2`

`1-cos^4(x) = 1- 1/4 = 3/4`

`8 sin^2xcos^2x != 1-cos^4x`