Complex Analysis Integrals

Question

I'm unsure how to apply what I've learned in complex analysis to the following question types:

$$ \int_{-\pi}^\pi \frac 1 {1 + \sin^2(\theta)}\,d\theta $$

and

$$ \int_{-\pi}^\pi \frac {\sin(n\theta)} {\sin(\theta)}\,d\theta. $$

Any help regarding where to look is greatly appreciated

Answer 1

Hint: $\displaystyle \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}\implies \sin\theta=\frac{z-z^{-1}}{2i},d\theta=\dfrac{dz}{iz}$