Mark the tick against the correct answer in the following:

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 $2 \tan ^{-1} x=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$

$\Rightarrow x=\sin \left(\sin ^{-1}\left(\frac{2\left(\frac{5}{8}\right)}{1+\left(\frac{5}{8}\right)^{2}}\right)=\sin \left(\sin ^{-1}\left(\frac{80}{89}\right)\right)=\frac{80}{89}\right.$

 

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