Question

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

• A. $\frac{12}{25}$
• B. $\frac{24}{25}$
• C. $\frac{1}{5}$
• D. None of these

Answer 1

The correct option is B

$\frac{24}{25}$

Explanation for the correct answer:

Simplifying the equations using trigonometric identities:

$$\begin{gathered} \Rightarrow \sin \left(2\times 2\times {\tan }^{-1}\left(\frac{1}{3}\right)\right)\left[\because 2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\right] \\ \Rightarrow \sin \left(2{\tan }^{-1}\left(\frac{{\displaystyle \frac{2}{3}}}{1-{\displaystyle \frac{1}{9}}}\right)\right) \\ \Rightarrow \sin \left(2{\tan }^{-1}\left(\frac{3}{4}\right)\right)\left[\because 2{\tan }^{-1}x={\sin }^{-1}\left(\frac{2x}{1-x^2}\right)\right] \\ \Rightarrow \sin \left[{\sin }^{-1}\left(\frac{2·{\displaystyle \frac{3}{4}}}{1+{\displaystyle \frac{9}{16}}}\right)\right] \\ \Rightarrow \left(\frac{3}{2}\right)\left(\frac{16}{25}\right) \\ \Rightarrow \frac{24}{25} \end{gathered}$$

Therefore, the correct answer is option (B).