Question

Prove the following
$\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }$ = 2 $csc\theta$

Answer 1

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