Inverse Trigonometric Functions

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

= tan ^-1 $\frac{sec-\theta }{tan\theta }$

tan ^-1 $\frac{\frac{1}{cos\theta }-1}{\frac{sin\theta }{cos\theta }}$ = tan ^-1 $\frac{\frac{1-cos\theta }{sin\theta }}{\frac{sin\theta }{cos\theta }}$ $=ta{n}^{+}\frac{1-cos\theta }{sin\theta }=ta{n}^{-1}\frac{2si{n}^2\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{2sin\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.cos\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

$=la{n}^{-1}\frac{sin\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{cos\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ { Ø cos 2Ø = 1 - sin²ØÞ 2 sin2 Ø = 1 - cos 2Ø => 2 sin2 $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ = 1 cos2 $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ = 1- cosØ }

$=ta{n}^{-1}\left(tan\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ Similar

$=\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ sin 2Ø = 2sinØ cosØ => sin $\frac{\theta }{2}$ = 2 sin $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ cos $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ => sin = 2 sin $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ cos $\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

$=\frac{ta{n}^{-1}x}{2}$