Question

Prove $\frac{\cot \theta +\csc \theta }{\cot \theta -\csc \theta }=\frac{\cos \theta +1}{\cos \theta -1}$

Answer 1

$LHS=\frac{\cot \theta +\text{cosec}\theta }{\cot \theta -\text{cosec}\theta }=\frac{\frac{\cos \theta }{\sin \theta }+\frac{1}{\sin \theta }}{\frac{\cos \theta }{\sin \theta }-\frac{1}{\sin \theta }}=\frac{\frac{\cos \theta +1}{\sin \theta }}{\frac{\cos \theta -1}{\sin \theta }}=\frac{\cos \theta +1}{\cos \theta -1}=RHS$

hence proved