Question
What is the value of $${\sec ^2}{\tan ^{ - 1}}\left( {\frac{5}{{11}}} \right)\,?$$
A.
$$\frac{{121}}{{96}}$$
B.
$$\frac{{211}}{{921}}$$
C.
$$\frac{{146}}{{121}}$$
D.
$$\frac{{267}}{{121}}$$
Answer :
$$\frac{{146}}{{121}}$$
Solution :
$$\eqalign{
& {\text{Let, }}{\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\frac{5}{{11}}} \right)} \right) \cr
& = 1 + {\tan ^2}\left( {{{\tan }^{ - 1}}\left( {\frac{5}{{11}}} \right)} \right)\left( {\because {{\sec }^2}\theta - {{\tan }^2}\theta = 1} \right) \cr
& = 1 + {\left[ {\tan \left( {{{\tan }^{ - 1}}\left( {\frac{5}{{11}}} \right)} \right)} \right]^2} = 1 + {\left( {\frac{5}{{11}}} \right)^2} \cr
& = 1 + \frac{{25}}{{121}} = \frac{{146}}{{121}} \cr} $$