If $\alpha$ is an acute angle, show that $\int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$
Solution 1:
Use $1 + \cos \alpha = 2\cos^2 (\frac{\alpha}{2})$
and $\sin \alpha = 2 \sin (\frac{\alpha}{2}) \cos (\frac{\alpha}{2})$
Solution 2:
Draw the right triangle to see that $\arctan(\cot(\alpha))=\frac{\pi}{2}-\alpha$