Question

Express $\sqrt{\frac{1-\sin A}{1+\sin A}}$ as sec A-tanA

Answer 1

$\sqrt{\frac{1-\sin A}{1+\sin A}}=\sec A-\tan A$

$LHS\to \sqrt{\frac{1-\sin A}{1+\sin A}}$

$=\sqrt{\frac{(1-\sin A)(1+\sin A)}{(1+\sin A)(1-\sin A)}}$

$=\frac{(1-\sin A)}{\sqrt{1-{\sin }^{2}A}}$

$=\frac{1-\sin A}{\sqrt{{\cos }^{2}A}}$

$=\frac{1-\sin A}{\cos A}$

$=\sec A-\tan A$

$=RHS.$