Q67 / 125 · BITSAT 2011 · Mathematics
The value of \(\sin \left(4 \tan ^{-1} \frac{1}{3}\right)-\cos \left(2 \tan ^{-1} \frac{1}{7}\right)\) is
- A\(3 / 7\)
- B\(7 / 8\)
- C\(8 / 21\)
- Correct answerDNone of these
Solution
\(\sin \left(4 \tan ^{-1} \frac{1}{3}\right)\)
\(\begin{aligned}
&= 2 \sin \left(2 \tan ^{-1} \frac{1}{3}\right) \cos \left(2 \tan ^{-1} \frac{1}{3}\right) \\
&=2 \sin \left(\tan ^{-1} \frac{3}{4}\right)=\cos \left(\tan ^{-1} \frac{3}{4}\right)=2 \cdot \frac{3}{5} \cdot \frac{4}{5}=\frac{24}{25} \\
& \cos \left(2 \tan ^{-1} \frac{1}{7}\right)=\cos \left(\tan ^{-1} \frac{7}{24}\right)=\frac{24}{25} .
\end{aligned}\)
The given expression \(=0\)
Hence, the correct answer is option: 'D'.