Question

Trig Identity help!

\[\sin x\left(\sin x +\cos x\right)^2\]

Answer 1

I need to find the Fourier series of this. So I am simply trying to write this as a sum of sines and cosines

Answer 2

wolfram alpha says that this is\[\frac{1}{2}\left(2\sin x+\cos x- \cos 3x\right)\]

Answer 3

\[f(x)=\sin x\left(\sin x +\cos x\right)^2\]\[\qquad=\sin x \left(1+2\sin x\cos x\right)\]\[\qquad = \sin x\left(1+\sin 2x\right)\]

Answer 4

but I think this approach leaves me at a dead end. I don't know any trig identities that could separate \(\sin x\sin 2x\) into two sums of sin/cosine only.

Answer 5

\[\frac{ \cos(-x)-\cos(x+2x) }{ 2}+\sin^3(x)+\sin(x)\cos^2(x)\]

Answer 6

how do you derive that identity?