Question

In Problems 9–23, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator.

${\cos }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right)$

Short answer

The exact value of the composite function is$\mathrm{\pi }.$

Step-by-step solution

Step 1. Given Information 

The given composite function is${\cos }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right).$

We have to find the exact value.

Step 2. Finding the value 

To find the exact value, let

$$\begin{gathered} x=\tan \frac{3\mathrm{\pi }}{4} \\ x=\tan \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right) \\ x=-\tan \frac{\mathrm{\pi }}{4}\left[\tan \left(\mathrm{\pi }-\mathrm{\theta }\right)=-\tan \theta \right] \end{gathered}$$

The value of $\tan \frac{\mathrm{\pi }}{4}=1$, $x=-1$

Step 3. Solve the composite fuction 

Now, substitute thexvalue in the given function.

Thus,

$$\begin{gathered} {\cos }^{-1}\left(\tan \frac{3\mathrm{\pi }}{4}\right) \\ ={\cos }^{-1}\left(-1\right) \\ ={\cos }^{-1}\left(\mathrm{cos\pi }\right)\left[\mathrm{cos\pi }=-1\right] \end{gathered}$$

By using the properties of an inverse function, $f\left(f^{-1}\left(x\right)\right)=\cos \left({\cos }^{-1}x\right)=xwhere-1\le x\le 1$

So,${\cos }^{-1}\left(\mathrm{cos\pi }\right)=\mathrm{\pi }$