TITLE: Interesting integral
QUESTION [24 upvotes]: Numerical evidence shows the validity of the following identity
$$\int\limits_0^z\frac{xdx}{\sin{x}\sqrt{\sin^2{z}-\sin^2{x}}}=\frac{\pi}{4\sin{z}}\ln{\frac{1+\sin{z}}{1-\sin{z}}},\tag{1}$$
if $0< z< \pi/2$.
How can it be proved? An indirect proof can be found in the paper http://link.springer.com/article/10.1134%2FS1547477113010044 (Potential of multiphoton exchange in the scattering of light charged particles of a heavy target, by  Yu.M. Bystritskiy, E.A. Kuraev and M.G. Shatnev). Formula (1) is equivalent to (12) from the paper where it is called "the marvelous identity".

REPLY [16 votes]: This integral is due to Lobachevskii. He gave it in more general form as follows

which can be found in his work "Application of imaginary geometry to certain integrals" (1836). Also see equation 3.842.7 in Gradsteyn and Ryzhik.
The integral in OP's post follows from this formula in the limit $~\alpha\to i\infty$.