Question

Show that: $\sqrt{\frac{1+\cos A}{1-\cos A}}=\text{cosec}A+\cot A$.

Answer 1

To prove: $\sqrt{\frac{1+cosA}{1-cosA}}=cosecA+cotA$

Consider $\sqrt{\frac{1+cosA}{1-cosA}}$

Rationalising

$\sqrt{\frac{1+cosA}{1-cosA}}=\sqrt{\frac{1+cosA}{1-cosA}\times \frac{1+cosA}{1+cosA}}$

$=\sqrt{\frac{(1+cosA{)}^2}{1-co{s}^{2}A}}$

$=\sqrt{\frac{(1+cosA{)}^2}{si{n}^{2}A}}$

$=\frac{1+cosA}{sinA}$

$=\frac{1}{sinA}+\frac{cosA}{sinA}$

$=cosecA+cotA$

$\therefore \sqrt{\frac{1+cosA}{1-cosA}}=cosecA+cotA$