If the function `f : [-1,1] to R` is continuous and even, then show that `int_(0)^(pi//2)f(cos2x)cosxdx=sqrt(2)int_(0)^(pi//4)f(sin2x)cosxdx`.

Explanation

To prove that

\[ \int_{0}^{\frac{\pi}{2}} f(\cos 2x) \cos x \, dx = \sqrt{2} \int_{0}^{\frac{\pi}{4}} f(\sin 2x) \cos x \, dx, \]

where \( f : [-1, 1] \to \mathbb{R} \) is continuous and even, we will start with the left-hand side (LHS) and manipulate it step by step.

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