Question

$(\text{cosec}A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$

Answer 1

$LHS=(\text{cosec}A-\sin A)(\text{sec}A-\cos A)=(\frac{1}{\sin A}-\sin A)(\frac{1}{\cos A}-\cos A)$

$=\left(\frac{1-{\sin }^{2}A}{\sin A}\right)\left(\frac{1-{\cos }^{2}A}{\cos A}\right)=\sin A\cos A=\frac{1}{2}\sin 2A$

$RHS=\frac{1}{\tan A+\cot A}=\frac{\tan A}{1+{\tan }^{2}A}=\frac{1}{2}\sin 2A$

$LHS=RHS$

Hence Proved