Question

The value of $\tan \left(\begin{array}{c}2{\tan }^{-1}\frac{1}{5}-\frac{\pi }{4}\end{array}\right)$ is

• A. $-\frac{7}{17}$
• B. $\frac{5}{16}$
• C. $\frac{5}{4}$
• D. $\frac{7}{17}$

Answer 1

The correct option is A $-\frac{7}{17}$

Solution:

$\tan (2{\tan }^{-1}\frac{1}{5}-\frac{\pi }{4})=$

$=\tan ({\tan }^{-1}\frac{5}{12}-{\tan }^{-1}1)$ ....... $[\because 2{\tan }^{-1}x={\tan }^{-1}\frac{2x}{1-x^2}]$

$=\tan \left({\tan }^{-1}\frac{-7}{17}\right)$ ...... $[\because {\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x-y}{1+xy}]$

$=\frac{-7}{17}$

Hence, A is the correct option.