252
Prove the following trigonometric identity :
$(\sin A - \csc A) (\cos A - \sec A) = \frac{1}{\tan A + \cot A}$
$(\sin A - \csc A) (\cos A - \sec A) = \frac{1}{\tan A + \cot A}$
Show SolutionHide Solution↓
Solution: LHS $= (\sin A - \frac{1}{\sin A}) (\cos A - \frac{1}{\cos A}) = \frac{\sin^2 A - 1}{\sin A} \times \frac{\cos^2 A - 1}{\cos A}$
$= \sin A \cos A = \frac{\sin A \cos A}{\sin^2 A + \cos^2 A}$
$= \frac{1}{\tan A + \cot A} = RHS$
$= \sin A \cos A = \frac{\sin A \cos A}{\sin^2 A + \cos^2 A}$
$= \frac{1}{\tan A + \cot A} = RHS$