Question

In Problems 79-90, find the exact value of each expression.

$\tan \left(2{\tan }^{-1}\frac{3}{4}\right)$

Short answer

The exact value is$\frac{24}{7}$

Step-by-step solution

Given information

Given expression$\tan \left(2{\tan }^{-1}\frac{3}{4}\right)$

Use the inverse rule and double angle rule and calculate

Here, $\tan \left({\tan }^{-1}x\right)=x$

$$\begin{gathered} \tan \left(2{\tan }^{-1}\frac{3}{4}\right)=a \\ {\tan }^{-1}\left(\frac{3}{4}\right)=\theta \\ a=\tan \left(2{\tan }^{-1}\frac{3}{4}\right) \\ a=\frac{2\tan \left({\tan }^{-1}\frac{3}{4}\right)}{1-{\tan }^2\left({\tan }^{-1}\frac{3}{4}\right)} \\ a=\frac{2\left(\frac{3}{4}\right)}{1-{\left(\frac{3}{4}\right)}^2} \\ a=\frac{24}{7} \end{gathered}$$