This question was posted on I&S
Prove the following
$$\int_0^\frac{\pi}{2} \cot^{-1}{\sqrt{1+\csc{\theta}}}\,\mathrm d\theta =\frac{\pi^2}{12}$$
The numerical value of the integral seems to agree with the answer.
Maybe someone could use that
$$1+\csc \theta = \csc(\theta) (\cos(\theta/2) + \sin(\theta/2))^2$$
I am sure there is a smart substitution or some trigonometric properties that I fail to see.