Question

In Problems 67–78,$f\left(x\right)=\sin x,-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2},g\left(x\right)=\cos x,0\le x\le \mathrm{\pi }\mathrm{and}\mathrm{h}\left(\mathrm{x}\right)=\mathrm{tanx},-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}$

Find the exact value of each composite function.

$g\left(h^{-1}\left(\frac{12}{5}\right)\right)$

Short answer

The required exact value of given expression is$\frac{5}{13}$.

Step-by-step solution

Step 1. Given information. 

The composite function$g\left(h^{-1}\left(\frac{12}{5}\right)\right)$.

Step 2. Here, let h-1125=x

So,

$$\begin{gathered} h\left(x\right)=\frac{12}{5} \\ \tan x=\frac{12}{5} \\ Giventhat-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2} \\ \sin ce,\tan x>0,so0<x<\frac{\mathrm{\pi }}{2} \\ Now, \\ secx=\sqrt{1+{\tan }^{2}x} \\ =\sqrt{1+{\left(\frac{12}{5}\right)}^2} \\ =\sqrt{1+\frac{144}{25}} \\ =\sqrt{\frac{169}{25}} \\ =\frac{13}{5} \end{gathered}$$

Step 3. Now, find gh-1125.

$$\begin{gathered} g\left(h^{-1}\left(\frac{12}{5}\right)\right)=g\left(x\right) \\ =\cos x \\ =\frac{1}{secx} \\ =\frac{1}{13/5} \\ =\frac{5}{13} \end{gathered}$$