Question

If f is an even function, then prove that \int\limits_0^{\pi /2} {f(\cos 2x)\cos x\,dx = \sqrt 2 \,\int\limits_0^{\pi /4} {f(\sin 2x)\cos x\,dx} }

Answer 1

@Dexter810
its for you if f(x) is an even function then prove that
\[ \int\limits\limits_0^{\pi /2} {f(\cos 2x)\cos x\,dx = \sqrt 2 \,\int\limits\limits_0^{\pi /4} {f(\sin 2x)\cos x\,dx} } \]