[latex] \frac{(tgt+ctgt)sint}{tgt} = sin^{-1} t[/latex]

[latex] \frac{(tgt+ctgt)\cdot sint}{tgt} = \frac{\left (\frac{sint}{cost}+\frac{cost}{sint}\right )\cdot sint}{\frac{sint}{cost}} =\frac{\frac{sin^2t+cos^2t}{sint\cdot cost}\cdot sint\cdot cost}{sint} =\\\\= \frac{1}{sint} =(sint)^{-1}[/latex]