Question

What is the value of \[{\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right)\]?
A. \[\dfrac{{121}}{{96}}\]
B. \[\dfrac{{217}}{{921}}\]
C. \[\dfrac{{146}}{{121}}\]
D. \[\dfrac{{267}}{{121}}\]

Answer 1

Hint: In this question, we have to find the value of \[{\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right)\]. For this first we will simplify it by assuming \[A = {\tan ^{ - 1}}\left( {\dfrac{5}{{11}}} \right)\]. Then using \[\tan {\tan ^{ - 1}}x = x\] we will further simplify it. Then we will use the identity \[{\sec ^2}x = 1 + {\tan ^2}x\] to rewrite the given expression and then we will substitute the obtained value of \[\tan {\tan ^{ - 1}}\left( {\dfrac{5}{{11}}} \right)\] and we will simplify it to find the result.

Complete step by step answer:
Here, we have to find the value of \[{\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right)\]. To solve this, we will assume \[A = {\tan ^{ - 1}}\left( {\dfrac{5}{{11}}} \right) - - - (1)\]. Taking \[\tan \] both the sides of \[(1)\], we get
\[ \Rightarrow \tan A = \tan {\tan ^{ - 1}}\left( {\dfrac{5}{{11}}} \right)\]
As we know that \[\tan {\tan ^{ - 1}}x = x\]. Using this, we get
\[ \Rightarrow \tan A = \dfrac{5}{{11}}\]
\[\Rightarrow \tan {\tan ^{ - 1}}\left( {\dfrac{5}{{11}}} \right) = \dfrac{5}{{11}} - - - (2)\]
Using the identity \[{\sec ^2}x = 1 + {\tan ^2}x\], we can write
\[ \Rightarrow {\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right) = 1 + {\tan ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right)\]

On rewriting, we get
\[ \Rightarrow {\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right) = 1 + {\left( {\tan \left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right)} \right)^2}\]
Using \[\left( 2 \right)\], we get
\[ \Rightarrow {\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right) = 1 + {\left( {\dfrac{5}{{11}}} \right)^2}\]
\[ \Rightarrow {\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right) = 1 + \dfrac{{25}}{{121}}\]
On simplifying, we get
\[ \therefore {\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right) = \dfrac{{146}}{{121}}\]
Therefore, the value of \[{\sec ^2}\left( {{{\tan }^{ - 1}}\left( {\dfrac{5}{{11}}} \right)} \right)\] is \[\dfrac{{146}}{{121}}\].

Hence, option C is correct.

Note: Here, we have used the trigonometric identity \[{\sec ^2}x = 1 + {\tan ^2}x\]. Trigonometric identities are equalities that involve trigonometric functions. An identity is an equation which is always true, no matter what values are substituted whereas an equation may not be true for some values that are substituted. There are many other identities that we can use according to the question to simplify the expression.