The value of
Question:

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

to :

1. $\frac{-181}{69}$

2. $\frac{220}{21}$

3. $\frac{-291}{76}$

4. $\frac{151}{63}$

Correct Option: , 2

Solution:

$\underbrace{\tan ^{-1} \frac{3}{5}+\tan ^{-1} \frac{3}{5}}_{x>0, y>0, x y<1}+\tan ^{-1} \frac{5}{12}$

$\tan ^{-1} \frac{\frac{6}{5}}{1-\frac{9}{25}}=\underbrace{\tan ^{-1} \frac{15}{8}+\tan ^{-1} \frac{5}{12}}_{x>0, y>0, x y<1}$

$\tan ^{-1} \frac{\frac{15}{8}+\frac{5}{12}}{1-\frac{15}{8} \cdot \frac{5}{12}}=\tan ^{-1} \frac{220}{21}$

$\tan \left(\tan ^{-1} \frac{220}{21}\right)=\frac{220}{21}$