Question

Prove that $\sqrt{\frac{\sin A+1}{1-\sin A}}+\sqrt{\frac{1-\sin A}{\sin A+1}}=2\sec A$

Answer 1

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

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

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

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

$=\frac{2}{\cos A}$

$=2\sec A$