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
Solution:
$\sqrt{\frac{1+\cos (A)}{1-\cos (A)}}=\sqrt{\frac{1+\cos (A)}{1-\cos (A)} \times \frac{1+\cos (A)}{1+\cos (A)}}=\frac{1+\cos (A)}{\sqrt{1-\cos ^{2}(A)}}$
$=\frac{1+\cos (A)}{\sqrt{\sin ^{2}(A)}}=\frac{1+\cos (A)}{\sin (A)}=\frac{1}{\sin (A)}+\frac{\cos (A)}{\sin (A)}$
$=\operatorname{cosec}(A)+\cot (A)$
Hence, the correct answer is option B.
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