Prove that $\int_{0}^{\pi/2}\frac{d\theta}{a+\sin^2{\theta}}=\frac{\pi}{2\sqrt{a(a+1))}}$ , if a>0

Question

Recently, I saw a similar question where you needed to prove that $$\int_{0}^{\infty}\frac{dx}{1+x^n}=\frac{\pi/n}{\sin{(\pi/n)}}$$.

The solution to that question is available at Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer..

However, I don't know if the same approach should be done or not. What should I do to solve this problem until I reach the final proof? In case there is the same problem already posted, please link me to it! Thank you for your attention.

Answer 1

$\textbf{Hint:}$ Assuming the simplest fix to a typo, we have

$$\frac{1}{a+\sin^2\theta} = \frac{1}{a\cos^2\theta+(a+1)\sin^2\theta} = \frac{1}{a+1}\cdot\frac{\sec^2\theta}{\frac{a}{a+1}+\tan^2\theta}$$