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.

$h\left(f^{-1}\left(-\frac{2}{5}\right)\right)$

Short answer

The required exact value of given expression is $\frac{-2}{\sqrt{21}}$.

Step-by-step solution

Step 1. Given information. 

The composite function$h\left(f^{-1}\left(-\frac{2}{5}\right)\right)$.

Step 2. Here, let f-1-25=x.

So,

$$\begin{gathered} f\left(x\right)=-\frac{2}{5} \\ \sin x=-\frac{2}{5} \\ \cos x=\sqrt{1-{\left(-\frac{2}{5}\right)}^2} \\ =\sqrt{1-\frac{4}{25}} \\ =\sqrt{\frac{21}{25}} \\ =\frac{\sqrt{21}}{5} \end{gathered}$$

Step 3. Now, find hf-1-25.

$$\begin{gathered} h\left(f^{-1}\left(-\frac{2}{5}\right)\right)=h\left(x\right) \\ =\tan x \\ =\frac{\sin x}{\cos x} \\ =\frac{-2/5}{\sqrt{21}/5} \\ =\frac{-2}{\sqrt{21}} \end{gathered}$$