Question

If ${\cot ^{ - 1}}\left( x \right) + {\tan ^{ - 1}}\left( 3 \right) = \dfrac{\pi }{2}$, then $x = $
A. $\dfrac{1}{3}$
B. $\dfrac{1}{4}$
C. $3$
D. $4$

Answer 1

Hint: In order to solve the equation, initiate with proving the resultant value ${\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \dfrac{\pi }{2}$ by following the appropriate steps then move ahead to compare the given values in the question with the obtained result. Solve the equations with the appropriate subtraction or addition method and get the value of $x$.

Complete step by step answer:
Considering ${\tan ^{ - 1}}x = p$ ……(1)
Multiplying both the sides by $\tan $, we get:
$\tan \left( {{{\tan }^{ - 1}}x} \right) = \tan p$
$ \Rightarrow x = \tan p$
Since, we know that $\tan p$ can be written in terms of $\cot $ as $\tan p = \cot \left( {\dfrac{\pi }{2} - p} \right)$.So, substituting this value in the above function, we get:
$ \Rightarrow x = \cot \left( {\dfrac{\pi }{2} - p} \right)$
Multiplying both the sides by ${\cot ^{ - 1}}$, we get:
$ \Rightarrow {\cot ^{ - 1}}x = {\cot ^{ - 1}}\left( {\cot \left( {\dfrac{\pi }{2} - p} \right)} \right)$
\[ \Rightarrow {\cot ^{ - 1}}x = \dfrac{\pi }{2} - p\] ……(2)

Adding equation 1 and 2 we get:
${\tan ^{ - 1}}x + {\cot ^{ - 1}}x = p + \dfrac{\pi }{2} - p$
Solving it further, we get:
$ \Rightarrow {\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \dfrac{\pi }{2}$ …..(3)
Subtracting both sides by ${\cot ^{ - 1}}\left( x \right)$:
$ \Rightarrow {\tan ^{ - 1}}x + {\cot ^{ - 1}}x - {\cot ^{ - 1}}x = \dfrac{\pi }{2} - {\cot ^{ - 1}}x$
$ \Rightarrow {\tan ^{ - 1}}x = \dfrac{\pi }{2} - {\cot ^{ - 1}}x$ ……(4)
Now, since we are given with the equation ${\cot ^{ - 1}}\left( x \right) + {\tan ^{ - 1}}\left( 3 \right) = \dfrac{\pi }{2}$.
Subtracting both the sides by ${\cot ^{ - 1}}\left( x \right)$,we get:
${\cot ^{ - 1}}\left( x \right) + {\tan ^{ - 1}}\left( 3 \right) - {\cot ^{ - 1}}\left( x \right) = \dfrac{\pi }{2} - {\cot ^{ - 1}}\left( x \right)$
$ \Rightarrow {\tan ^{ - 1}}\left( 3 \right) = \dfrac{\pi }{2} - {\cot ^{ - 1}}\left( x \right)$
From equation 4, we can write it as:
$ \Rightarrow {\tan ^{ - 1}}\left( 3 \right) = {\tan ^{ - 1}}\left( x \right)$
Multiplying both the sides by $\tan $, we get:
$ \Rightarrow \tan \left( {{{\tan }^{ - 1}}\left( 3 \right)} \right) = \tan \left( {{{\tan }^{ - 1}}\left( x \right)} \right)$
$ \Rightarrow 3 = x$
$ \therefore x = 3$
Therefore, the value of $x = 3$.

Hence, option C is correct.

Note:We could have alternatively directly solved the equation by comparing the given equation with the obtained equation 3, which would have resulted in the value of $x$ to be $3$. It’s important to prove the results before applying direct values otherwise it may lead to errors. $\tan p$ is written in terms of $\cot $ as $\tan p = \cot \left( {\dfrac{\pi }{2} - p} \right)$ because $\dfrac{\pi }{2}$ is an odd function, so it changes the value outside the bracket into its reverse one and obtains a positive value because $\dfrac{\pi }{2} - p$ lies in the first quadrant and in first quadrant all the values are positive. Similarly, we can write cot in terms of tan.