Ideas about integral

Question

The problem asks to compute the integral $I := \displaystyle\int_0^\pi \sin^{2n}(t) \cos(2nt) dt$ ($n$, a natural number)

Testing with wolframalpha suggests that $I = = (-1)^n \dfrac{\pi}{4^n}$

Any ideas on how to prove this ?

(writing $\cos(2nt) = \Re(e^{i2nt})$ doesn't seem to simplify the computation)

Thanks