# How to prove this trigonometric integral?

$$\displaystyle \int_{-\pi/4}^{\pi/4} {{\left(\dfrac{\cos x - \sin x}{\cos x + \sin x}\right)}^{\cos(2t)} \ dx} = \frac{\pi}{2 \sin(\pi \cos^2 t)}$$

I could simplify it to

$\displaystyle \int_0^1 {\left(t^n + \frac{1}{t^n}\right) \ \frac{dt}{1+t^2}}, \ n = \cos 2t$

From here, I can think of expanding into sums but that doesn't seem a good option. Also, getting back to trigonometric form is also an option but it would get us to reduction formula which will be messy.

What is a straight, neat and easy approach to solve it?

• Can you use the residue theorem? – jim May 7 '16 at 8:29