Question

The value of $\tan (2{\tan }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right))$ is equal to:

• A. $\frac{220}{21}$
• B. $\frac{151}{63}$
• C. $\frac{-181}{69}$
• D. $\frac{-291}{76}$

Answer 1

The correct option is A $\frac{220}{21}$

$2{\tan }^{-1}\left(\frac{3}{5}\right)={\tan }^{-1}\left(\frac{\frac{6}{5}}{1-\frac{9}{25}}\right)={\tan }^{-1}\left(\frac{\frac{6}{5}}{\frac{16}{25}}\right)={\tan }^{-1}\frac{15}{8}$

$\therefore 2{\tan }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right)={\tan }^{-1}\left(\frac{15}{8}\right)+{\tan }^{-1}\left(\frac{5}{12}\right)$

$={\tan }^{-1}\left(\frac{\frac{15}{8}+\frac{5}{12}}{1-\frac{15}{8}.\frac{5}{12}}\right)$

$={\tan }^{-1}\left(\frac{180+40}{21}\right)={\tan }^{-1}\left(\frac{220}{21}\right)$
$\therefore \tan (2{\tan }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right))=\frac{220}{21}$