If $f$ is continuous on $[0,1]$, how can I show that $$\int\_0^{n\pi}f(\cos^2(x))\,dx=n\int\_0^{\pi}f(\cos^2(x))\,dx?$$

I know how to show that both integrals exist. I believe that $n$ is a multiple of $\frac12$. I thought that the identity $\cos^2(x)=\frac{\cos(2x)+1}{2}$ would be useful, but maybe not. I think I should use some change of variable.