Question

$4{\tan }^{-1}\left(\frac{1}{5}\right)-{\tan }^{-1}\left(\frac{1}{239}\right)=?$

• A. $\mathrm{\pi }$
• B. $\frac{\mathrm{\pi }}{2}$
• C. $\frac{\mathrm{\pi }}{3}$
• D. $\frac{\mathrm{\pi }}{4}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{4}$

Explanation for the correct option:

Step 1: Expand the first term of the expression.

Use inverse trigonometric identity to expand the expression.

$\begin{array}{rcl}4{\tan }^{-1}\left(\frac{1}{5}\right)& =& 2\left(2t{\mathrm{an}}^{-1}\left(\frac{1}{5}\right)\right)\\ & =& 2t{\mathrm{an}}^{-1}\left(\frac{\frac{2}{5}}{1-{\displaystyle \frac{1}{25}}}\right)\\ & =& 2t{\mathrm{an}}^{-1}\left(\frac{10}{24}\right)\end{array}$ $\left[\because 2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\right]$

Step 2: Expand the obtained expression.

Use inverse trigonometric identity to expand the expression.

$\begin{array}{rcl}2t{\mathrm{an}}^{-1}\left(\frac{10}{24}\right)& =& {\tan }^{-1}\frac{2·\left({\displaystyle \frac{10}{24}}\right)}{1-{\displaystyle \frac{100}{576}}}\\ & =& {\tan }^{-1}\frac{120}{119}\end{array}$ $\left[\because 2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\right]$

Step 3: Put the obtained value in the given expression and simplify.

Add the obtained expression using inverse trigonometric identity.

$\begin{array}{rcl}4{\tan }^{-1}\left(\frac{1}{5}\right)-{\tan }^{-1}\left(\frac{1}{239}\right)& =& {\tan }^{-1}\frac{120}{119}-{\tan }^{-1}\left(\frac{1}{239}\right)\\ & =& {\tan }^{-1}\left(\frac{\frac{120}{119}-\frac{1}{239}}{1+\left(\frac{120}{119}\right)\left(\frac{1}{239}\right)}\right)\\ & =& {\tan }^{-1}\left(\frac{120·239-119}{119·239+120}\right)\\ & =& {\tan }^{-1}\left(\frac{28561}{28561}\right)\\ & =& {\tan }^{-1}1\end{array}$ $\left[\because {\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right)\right]$

So, $\begin{array}{rcl}4{\tan }^{-1}\left(\frac{1}{5}\right)-{\tan }^{-1}\left(\frac{1}{239}\right)& =& {\tan }^{-1}1\\ & =& \frac{\mathrm{\pi }}{4}\end{array}$

Hence, option (D) is the correct answer.