Question

Find the exact value of the expression. Express angles in radians.

$\tan \left({\tan }^{-1}\frac{7}{3}\right)$

Short answer

The exact value of the given expression is $\frac{7}{3}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\tan \left({\tan }^{-1}\frac{7}{3}\right)$

For the tangent function and its inverse property,

$$\begin{gathered} f\left(f^{-1}\left(x\right)\right)=\tan \left({\tan }^{-1}x\right) \\ =x \end{gathered}$$

Where x is the angle between $-\infty \le x\le \infty$.

Step 2. Solve the equation.

Consider the given question,

$\tan \left({\tan }^{-1}\frac{7}{3}\right)$

Using the tangent function and inverse property,

$$\begin{gathered} =\tan \left({\tan }^{-1}\frac{7}{3}\right) \\ =\frac{7}{3} \end{gathered}$$

Therefore, the exact value of$\tan \left({\tan }^{-1}\frac{7}{3}\right)$is$\frac{7}{3}$.