Question

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

Answer 1

To prove$\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\cos ec\theta$.

consider L.H.S.:

$\begin{array}{rcl}\mathrm{LHS}& =& \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }\\ & =& \frac{{\sin }^2\theta +{(1+\cos \theta )}^2}{\sin \theta ·(1+\cos \theta )}\\ & =& \frac{{\sin }^2\theta +{\cos }^2\theta +2\cos \theta +1}{\sin \theta ·(1+\cos \theta )}\left[\because {(a+b)}^2=a^2+2ab+b^2\right]\\ & =& \frac{1+2\cos \theta +1}{\sin \theta ·(1+\cos \theta )}\left[\because {\sin }^2\theta +{\cos }^2\theta =1\right]\\ & =& \frac{2(1+\cos \theta )}{\sin \theta ·(1+\cos \theta )}\\ & =& \frac{2}{\sin \theta }\\ & =& 2\cos ec\theta \left[\because \frac{1}{\sin \theta }=\cos ec\theta \right]\\ & =& \mathrm{RHS}\end{array}$

Thus,$\mathrm{LHS}=\mathrm{RHS}$

Hence,$\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\cos ec\theta$ is proved.