How to Verify the Complex Integral Equals π/(1+n)?

[Tony1]

How to prove this integral,

$$\int_{0}^{2\pi}\mathrm dt{\sin t\over \sin t+ i\sqrt{n+\cos^2 t}}={\pi\over 1+n}$$

$n \ne -1$

$i=\sqrt{-1}$

 

[Opalg]

How to prove this integral,

$$\int_{0}^{2\pi}\mathrm dt{\sin t\over \sin t+ i\sqrt{n+\cos^2 t}}={\pi\over 1+n}$$

$n \ne -1$

$i=\sqrt{-1}$

Spoiler

Rationalise the fraction: \(\displaystyle \frac{\sin t}{ \sin t+ i\sqrt{n+\cos^2 t}} = \frac{\sin t\bigl( \sin t - i\sqrt{n+\cos^2 t}\bigr)}{1+n}.\)