Question

If $\tan \theta =\frac{1}{\sqrt{15}}$ and $0<\theta <\frac{\pi }{2}$, then find the value of $\frac{cs{c}^2\theta -{\sec }^2\theta }{cs{c}^2\theta +{\sec }^2\theta }$

• A. $\frac{5}{6}$
• B. $\frac{6}{7}$
• C. $\frac{7}{8}$
• D. $\frac{8}{9}$

Answer 1

The correct option is C $\frac{7}{8}$

Given $\tan \theta =\frac{1}{\sqrt{15}}⟹\cot \theta =\sqrt{15}$

$\frac{{\text{csc}}^2\theta -{\text{sec}}^2\theta }{{\text{csc}}^2\theta +{\text{sec}}^2\theta }=\frac{1+{\cot }^2\theta -1-{\tan }^2\theta }{1+{\cot }^2\theta +1+{\tan }^2\theta }=\frac{15-\frac{1}{15}}{2+15+\frac{1}{15}}=\frac{224}{256}=\frac{7}{8}$