Question

Question #73def

Answer 1

Apply the double-angle formulae to prove this.

Explanation:

The double-angle formula for `sin, cos, tan` are:

`sin2x=2sinxcosx`
`cos2x=cos^2x-sin^2x=1-2sin^2x=2cos^2x-1`
`tan2x=(2tanx)/(1-tan^2x)`.

Then, let's prove the given equation.

(LHS)`=(1-cos8x)/8=(1-cos(2*4x))/8`
`=(1-(1-2sin^2 4x))/8`
`=(2sin^2 4x)/8=(sin^2 4x)/4`

(RHS)`=sin^2 2x cos^2 2x`
`=(1/2*2sin2xcos2x)^2`
`=(1/2sin(2*2x))^2`
`=(1/2sin4x)^2=(sin^2 4x)/4`

In this way, (LHS)`=`(RHS) is proven.