Prove that $\langle\sin\theta\rangle=\langle|\cos\theta|\rangle$

Question

Prove that $\langle\sin\theta\rangle=\langle|\cos\theta|\rangle$ where $\langle x\rangle$ represents the mean of $x$ and $\theta$ is a uniform random variable that takes values between $0$ and $\pi$.

I know how to prove this the long way, finding the value of each side, which happens to be $\displaystyle\frac{2}{\pi}$. But I think there should be a quick trick.

Answer 1

In $[0,\pi]$, $\sin(\theta)=\left|\cos\left(\dfrac\pi2-\theta\right)\right|$ and by a change of variable the integrals coincide.