Question

Prove that :-
$\frac{1+\cos \theta }{\sin \theta }+\frac{\sin \theta }{1+\cos \theta }=2.\cos ec\theta$

Answer 1

L.H.S

$\frac{1+\cos \theta }{\sin \theta }+\frac{\sin \theta }{1+\cos \theta }$

$=\frac{{(1+\cos \theta )}^2+{\sin }^2\theta }{\sin \theta (1+\cos \theta )}$

$=\frac{1+{\cos }^2\theta +2\cos \theta +{\sin }^2\theta }{\sin \theta (1+\cos \theta )}$

$=\frac{1+{\cos }^2\theta +{\sin }^2\theta +2\cos \theta }{\sin \theta (1+\cos \theta )}(Since:{\sin }^{2}x+{\cos }^{2}x=1)$

$=\frac{1+1+2\cos \theta }{\sin \theta (1+\cos \theta )}$

$=\frac{2+2\cos \theta }{\sin \theta (1+\cos \theta )}$

$=\frac{2(1+\cos \theta )}{\sin \theta (1+\cos \theta )}$

$=\frac{2}{\sin \theta }$

$=2\csc \theta$

Hence, proved.