(cosec θ − cot θ)2 = ?

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) $\frac{1-\cos \theta}{1+\cos \theta}$

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

$=\left(\frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\right)^{2}$

$=\left(\frac{1-\cos \theta}{\sin \theta}\right)^{2}$

$=\frac{(1-\cos \theta)^{2}}{\sin ^{2} \theta}$

$=\frac{(1-\cos \theta)^{2}}{\left(1-\cos ^{2} \theta\right)}$

$=\frac{(1-\cos \theta)^{2}}{(1+\cos \theta)(1-\cos \theta)}$

$=\frac{(1-\cos \theta)}{(1+\cos \theta)}$

 

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