Question

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

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

Short answer

The exact value is$\frac{25}{7}$

Step-by-step solution

Given information

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

Rewriting using inverse rule and calculating secant

Rewriting, we get

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

Using double angle formula and calculating

Calculating, we get

$$\begin{gathered} \sec \left(2{\tan }^{-1}\frac{3}{4}\right)=\frac{1}{2{\cos }^2\theta -1} \\ \cos \theta =\frac{4}{5} \\ \sec \left(2{\tan }^{-1}\frac{3}{4}\right)=\frac{1}{2{\left(\frac{4}{5}\right)}^2-1} \\ \sec \left(2{\tan }^{-1}\frac{3}{4}\right)=\frac{1}{\frac{32}{25}-1} \\ \sec \left(2{\tan }^{-1}\frac{3}{4}\right)=\frac{25}{32-25} \\ \sec \left(2{\tan }^{-1}\frac{3}{4}\right)=\frac{25}{7} \end{gathered}$$