Question

The value of $\sin \left(4{\tan }^{-1}\frac{1}{3}\right)-\cos \left(2{\tan }^{-1}\frac{1}{7}\right)$ is

• A. $\frac{3}{7}$
• B. 0
• C. $\frac{8}{21}$
• D. $\frac{24}{25}$

Answer 1

Answer: (B)

$\sin \left(4{\tan }^{-1}\frac{1}{3}\right)=2\sin \left(2{\tan }^{-1}\frac{1}{3}\right)\cos \left(2{\tan }^{-1}\frac{1}{3}\right)$
$=2\sin \left({\tan }^{-1}\frac{3}{4}\right)\cos \left({\tan }^{-1}\frac{3}{4}\right)=2\cdot \frac{3}{5}\cdot \frac{4}{5}=\frac{24}{25}$
$\cos \left(2{\tan }^{-1}\frac{1}{7}\right)=\cos \left({\tan }^{-1}\frac{7}{24}\right)=\frac{24}{25}$
The given expression $=0$