The expression in the image is:

$\tan\left(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)$

Step 1: Solve $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$

$\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$ means we are looking for the angle $\theta$ such that:

$\sin(\theta) = \frac{\sqrt{3}}{2}$

From trigonometric values, we know that $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$. Therefore,

$\theta = \frac{\pi}{3}$

Step 2: Apply the $\tan$ function

Now we need to find:

$\tan\left(\frac{\pi}{3}\right)$

From standard trigonometric values, we know that:

$\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$

Final Answer:

The exact value of the given expression is:

$\sqrt{3}$

Would you like more details or further explanations?

Here are some related questions:

1. What is the definition of the inverse sine function?
2. How are trigonometric values derived from the unit circle?
3. What are other common values for $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$?
4. How can you verify trigonometric values using a calculator?
5. What are some common applications of inverse trigonometric functions in geometry?

Tip: It helps to memorize the key angles (like $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$) and their sine, cosine, and tangent values for quick reference!