Question

How do you prove `sin4x = 4sinxcos^3x - 4sin^3xcosx`?

Answer 1

I'll start from the double angle identities:

`cos 2theta = cos^2 theta - sin^2 theta`

`sin 2theta = 2sin theta cos theta`

Then:

`sin 4x = 2sin 2x cos 2x`

`=2(2 sin x cos x)(cos ^2x - sin^2 x)`

`= 2(2sin x cos x cos^2 x - 2sinx cos x sin^2 x)`

`=4sin x cos^3 x - 4 sin^3 x cos x`