I have the following integral,
$I(t)=\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}},$
where $t>4$ is a real parameter. I know from messing around numerically and playing with Mathematica that
$I(t)=\frac{4}{t}K\left(\frac{16}{t^2}\right),$
where $K$ is the complete elliptic integral of the first kind with parameter $m=k^2=16/t^2$. However, I seek proof of that fact. I have tried a handful of changes of variables which didn't get the job done, and I've searched tables of integrals without finding this integrand or similar. Any suggestions or hints would be appreciated.