Question:
(cosec θ − cot θ)2 = ?
(a) $\frac{1+\cos \theta}{1-\cos \theta}$
(b) $\frac{1-\cos \theta}{1+\cos \theta}$
(c) $\frac{1+\sin \theta}{1-\sin \theta}$
(d) $\frac{1-\sin \theta}{1+\sin \theta}$
Solution:
(b) $\frac{1-\cos \theta}{1+\cos \theta}$
$(\operatorname{cosec} \theta-\cot \theta)^{2}$
$=\left(\frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\right)^{2}$
$=\left(\frac{1-\cos \theta}{\sin \theta}\right)^{2}$
$=\frac{(1-\cos \theta)^{2}}{\sin ^{2} \theta}$
$=\frac{(1-\cos \theta)^{2}}{\left(1-\cos ^{2} \theta\right)}$
$=\frac{(1-\cos \theta)^{2}}{(1+\cos \theta)(1-\cos \theta)}$
$=\frac{(1-\cos \theta)}{(1+\cos \theta)}$