Question

use sum, difference, product, or half-angle formulas to find the exact value of each expression

$\tan \left(2{\sin }^{-1}\right(\frac{6}{11}\left)\right)$

Short answer

$\tan \left(2{\sin }^{-1}\right(\frac{6}{11}\left)\right)=\frac{12\sqrt{85}}{49}$

Step-by-step solution

Step 1.Given expression

$\tan \left(2{\sin }^{-1}\right(\frac{6}{11}\left)\right)$

Step 2. Solving 

let :

${\sin }^{-1}\frac{6}{11}=\theta$

we need to find

$\tan 2\theta$

$$\begin{gathered} {\sin }^{-1}\frac{6}{11}=\theta \\ \sin \theta =\frac{6}{11} \end{gathered}$$

Step 3. Solving for tan 2θ

$$\begin{gathered} \tan 2\theta =\frac{\sin 2\theta }{\cos 2\theta } \\ \tan 2\theta =\frac{2\sin \theta \cos \theta }{1-2{\sin }^2\theta } \\ \tan 2\theta =\frac{2\sin \theta \sqrt{1-{\sin }^2\theta }}{1-2{\sin }^2\theta } \\ \tan 2\theta =\frac{2\times {\displaystyle \frac{6}{11}}\sqrt{1-{\left({\displaystyle \frac{6}{11}}\right)}^2}}{1-2\times {\left({\displaystyle \frac{6}{11}}\right)}^2} \\ \tan 2\theta =\frac{2\times {\displaystyle \frac{6}{11}}{\displaystyle \frac{\sqrt{85}}{11}}}{{\displaystyle \frac{49}{121}}} \\ \tan 2\theta =\frac{12\sqrt{85}}{49} \end{gathered}$$