Question

Prove that $\sqrt{\frac{1+\cos A}{1-\cos A}}$ = (cosec A + cot A).

Answer 1

$\sqrt{\frac{1+\cos A}{1-\cos A}}$= (cosec A + cot A).

$$\begin{gathered} \text{LHS =}\sqrt{\frac{1+\cos A}{1-\cos A}} \\ \text{Multiplying the numerator and denominator by}(1+\cos A),\mathrm{we}\mathrm{have}: \\ \sqrt{\frac{{(1+\cos A)}^2}{(1-\cos A)(1+\cos A)}} \\ =\sqrt{\frac{{(1+\cos A)}^2}{1-{\cos }^{2}A}} \\ =\frac{1+\cos A}{\sqrt{{\sin }^{2}A}} \\ =\frac{1+\cos A}{\sin A} \\ =\frac{1}{\sin A}+\frac{\cos A}{\sin A} \\ =\cos \text{ec}A+\cot A=\text{RHS} \\ \text{Hence proved}\text{.} \end{gathered}$$