The value of $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}$ is
(a) $\cot \theta-\operatorname{cosec} \theta$
(b) $\operatorname{cosec} \theta+\cot \theta$
(c) $\operatorname{cosec}^{2} \theta+\cot ^{2} \theta$
(d) $(\cot \theta+\operatorname{cosec} \theta)^{2}$
The given expression is $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}$.
Multiplying both the numerator and denominator under the root by $(1+\cos \theta)$, we have
$\sqrt{\frac{(1+\cos \theta)(1+\cos \theta)}{(1+\cos \theta)(1-\cos \theta)}}$
$=\sqrt{\frac{(1+\cos \theta)^{2}}{\left(1-\cos ^{2} \theta\right)}}$
$=\sqrt{\frac{(1+\cos \theta)^{2}}{\sin ^{2} \theta}}$
$=\frac{1+\cos \theta}{\sin \theta}$
$=\frac{1}{\sin \theta}+\frac{\cos \theta}{\sin \theta}$
$=\operatorname{cosec} \theta+\cot \theta$
Therefore, the correct choice is (b).
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