Question

Prove:$\frac{\cot \theta -\cos \theta }{\cot \theta +\cos \theta }=\frac{\mathrm{cosec}\theta -1}{\mathrm{cosec}\theta +1}$

Answer 1

L.H.S

$\Rightarrow \frac{\cot \theta -\cos \theta }{\cot \theta +\cos \theta }$

$\Rightarrow \frac{\frac{\cos \theta }{\sin \theta }-\cos \theta }{\frac{\cos \theta }{\sin \theta }+\cos \theta }$

$\Rightarrow \frac{\frac{1}{\sin \theta }-1}{\frac{1}{\sin \theta }+1}$

$\Rightarrow \frac{\csc \theta -1}{\csc \theta +1}$

R.H.S

Hence, proved.