Prove that (cosec 2 θ – sec 2 θ) / (cosec 2 θ + sec 2 θ) = 3/4

Q) Prove that $\frac {\cos ec ^2 \theta - \sec ^2 \theta}{\cos ec ^2 \theta + \sec ^2 \theta} = \frac{3}{4}$ , if $\tan \theta = \frac{1}{\sqrt 7}$ .

Ans: Let’s start from LHS:

LHS = $\frac {\cos ec ^2 \theta - \sec ^2 \theta}{\cos ec ^2 \theta + \sec ^2 \theta}$

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

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

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