Question

solve this question ​

Answer 1

$\bf{\Huge{\boxed{\tt{\purple{ANSWER\::}}}}}$

$\bf{\Large{\underline{\bf{Given\::}}}}$

$\sf{\large{\sqrt{\frac{sec\theta-1}{sec\theta+1} } +\sqrt{\frac{sec\thtea+1}{sec\theta-1} } \:=\:2cosec\theta}}}$

$\bf{\Large{\underline{\rm{\red{Proof\::}}}}}$

$\hookrightarrow\sf{\underline{\bigstar{Take\:L.H.S.}}}$

$\hookrightarrow\sf{\large{\sqrt{\frac{sec\theta-1}{sec\theta+1} } +\sqrt{\frac{sec\thtea+1}{sec\theta-1} }}}$

$\hookrightarrow\sf{\large{\frac{(\sqrt{(sec\theta-1)^{2} }+\sqrt{(sec\theta+1)^{2} } }{\sqrt{sec\theta+1} \sqrt{sec\theta-1}}}}$

$\hookrightarrow\sf{\large{\frac{sec\theta-1+sec\theta+1}{\sqrt{(sec^{2} \theta-1)} } }}$

$\hookrightarrow\sf{\large{\frac{sec\theta\cancel{-1}+sec\theta\cancel{+1}}{\sqrt{(sec^{2} \theta-1)} } }}$

$\hookrightarrow{\sf{\large{\frac{2sec\theta}{tan\theta} }}}$

$\hookrightarrow\sf{\large{\frac{2*\cancel{cos\theta}}{\cancel{cos\theta}*sin\theta} }}$

$\hookrightarrow\sf{\large{\frac{2}{sin\theta} }}}$

$\hookrightarrow\sf{\large{\boxed{\pink{2\:cosec\theta}}}}$

Hence,

Proved.