# Calculate double integral for a segment of a sphere

I have one problem and the exact answer for it. But did not figure out the solution. $$A_{P-A} = \large R^2\int_{\theta=\frac{\pi}{2}-\beta}^{\alpha_{eq}}\int_{\varphi=\arcsin\left(\frac{1}{\tan\beta\tan\theta}\right)}^{\pi-\arcsin\left(\frac{1}{\tan\beta\tan\theta}\right)}\sin\theta\ d\theta\ d\varphi$$

I have tried to do that with mathematica but it do not provide the same solution rather it provides very big non realistic solution. I can solve up to the following but do not know how to proceed later.

Can anyone help me to continue?