Question

Check whether $2{\tan }^{-1}\frac{1}{5}+{\sec }^{-1}\frac{5\sqrt{2}}{7}+2{\tan }^{-1}\frac{1}{8}=\frac{\pi }{4}$ is true/false?

• A. True
• B. False

Answer 1

The correct option is A True

$2{\tan }^{-1}\frac{1}{5}+{\sec }^{-1}\frac{5\sqrt{2}}{7}+2{\tan }^{-1}\frac{1}{8}$

$=2{\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\sqrt{{\left(\frac{5\sqrt{2}}{7}\right)}^2-1}+2{\tan }^{-1}\frac{1}{8}$

$=2{\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{7}+2{\tan }^{-1}\frac{1}{8}$

$=2({\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{8})+{\tan }^{-1}\frac{1}{7}$

$=2{\tan }^{-1}\frac{\frac{1}{5}+\frac{1}{8}}{1-\frac{1}{5}\times \frac{1}{8}}+{\tan }^{-1}\frac{1}{7}$

$=2{\tan }^{-1}\frac{13}{39}+{\tan }^{-1}\frac{1}{7}$

$=2{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{7}$

$={\tan }^{-1}\left(\frac{\frac{2}{3}}{1-\frac{1}{9}}\right)+{\tan }^{-1}\frac{1}{7}$

$={\tan }^{-1}\frac{3}{4}+{\tan }^{-1}\frac{1}{7}$

$={\tan }^{-1}\left(\frac{\frac{3}{4}+\frac{1}{7}}{1-\frac{3}{4}\times \frac{1}{7}}\right)$

$={\tan }^{-1}\frac{21+4}{28-3}$

$={\tan }^{-1}\frac{25}{25}$

$={\tan }^{-1}1=\frac{\pi }{4}$