Question

$\sqrt{\frac{1+\cos A}{1-\cos A}}=?$
(a) cosec A – cot A
(b) cosec A + cot A
(c) cosec A cot A
(d) none of these

Answer 1

$$\begin{gathered} \sqrt{\frac{1+\cos \left(A\right)}{1-\cos \left(A\right)}}=\sqrt{\frac{1+\cos \left(A\right)}{1-\cos \left(A\right)}\times \frac{1+\cos \left(A\right)}{1+\cos \left(A\right)}}=\frac{1+\cos \left(A\right)}{\sqrt{1-{\cos }^2\left(A\right)}} \\ =\frac{1+\cos \left(A\right)}{\sqrt{{\sin }^2\left(A\right)}}=\frac{1+\cos \left(A\right)}{\sin \left(A\right)}=\frac{1}{\sin \left(A\right)}+\frac{\cos \left(A\right)}{\sin \left(A\right)} \\ =\mathrm{cosec}\left(A\right)+\cot \left(A\right). \end{gathered}$$
Hence, the correct answer is option B.