Question:
Mark the tick against the correct answer in the following:
$\sin \left[2 \tan ^{-1} \frac{5}{8}\right]$
A. $\frac{25}{64}$
B. $\frac{80}{89}$
C. $\frac{75}{128}$
D. none of these
Solution:
To Find: The value of $\sin \left(2 \tan ^{-1} \frac{5}{8}\right)$
Let, $x=\sin \left(2 \tan ^{-1} \frac{5}{8}\right)$
We know that $2 \tan ^{-1} x=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$
$\Rightarrow x=\sin \left(\sin ^{-1}\left(\frac{2\left(\frac{5}{8}\right)}{1+\left(\frac{5}{8}\right)^{2}}\right)=\sin \left(\sin ^{-1}\left(\frac{80}{89}\right)\right)=\frac{80}{89}\right.$