For what $n$ the set $\{\sin x, \cos x, (\sin x)^2, (\cos x)^2,..., (\sin x)^n, (\cos x)^n\}$ is linearly independent?

Question

Under what condition of $n$ the following set $\{\sin x, \cos x, \sin^2x, \cos^2x,..., \sin^nx, \cos^n x\}$ is linearly independent? I tried to replace n=1,2,3 but I haven't get the general result. Could you please help me?

Answer 1

If you can prove the set as linearly independent for $n\le3$, you can show the set is not linearly independent for $n>3$. JJacquelin's comment gave me the idea. We have

$$\cos^4x-\sin^4x=(\cos^2x-\sin^2x)(\cos^2x+\sin^2x)$$ $$\cos^4x-\sin^4x-\cos^2x+\sin^2x=0$$