Question

Show that $(co\sec A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$

Answer 1

$LHS=(\text{cosec}A-\sin A)(\text{sec}A-\cos A)=(\frac{1-{\sin }^{2}A}{\sin A})(\frac{1-{\cos }^{2}A}{\cos A})=\frac{{\sin }^{2}A{\cos }^{2}A}{\sin A\cos A}=\sin A\cos A$

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

$LHS=RHS$

Hence Proved