Question

$frac{(tgt+ctgt)sint}{tgt} = sin^{-1} t$

Answer 1

$frac{(tgt+ctgt)cdot sint}{tgt} = frac{left (frac{sint}{cost}+frac{cost}{sint}right )cdot sint}{frac{sint}{cost}} =frac{frac{sin^2t+cos^2t}{sintcdot cost}cdot sintcdot cost}{sint} =\\= frac{1}{sint} =(sint)^{-1}$