# The value of 1+cos θ1−cos θ−−−−−−√ is

Question:

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) $(\cot \theta+\operatorname{cosec} \theta)^{2}$

Solution:

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).