Let's break down and solve the expression step by step:

You have: $\sin \left( \cos^{-1} \left( \frac{\sqrt{2}}{2} \right) + \tan^{-1} \left( \sqrt{3} \right) \right)$

Step 1: Understand the individual components

$\cos^{-1} \left( \frac{\sqrt{2}}{2} \right)$

This is the angle $\theta$ such that: $\cos(\theta) = \frac{\sqrt{2}}{2}$ We know that $\cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$, so: $\cos^{-1} \left( \frac{\sqrt{2}}{2} \right) = \frac{\pi}{4}$

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

This is the angle $\phi$ such that: $\tan(\phi) = \sqrt{3}$ We know that $\tan \left( \frac{\pi}{3} \right) = \sqrt{3}$, so: $\tan^{-1} \left( \sqrt{3} \right) = \frac{\pi}{3}$

Step 2: Substitute the values into the expression

Now we substitute the values for the inverse trigonometric functions: $\sin \left( \frac{\pi}{4} + \frac{\pi}{3} \right)$

Step 3: Simplify the angle

Let's add the angles: $\frac{\pi}{4} + \frac{\pi}{3} = \frac{3\pi}{12} + \frac{4\pi}{12} = \frac{7\pi}{12}$

Step 4: Find the sine of the resulting angle

We need to calculate: $\sin \left( \frac{7\pi}{12} \right)$ We can use a sine addition formula: $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ where $a = \frac{\pi}{4}$ and $b = \frac{\pi}{3}$.

Substitute the known values for $\sin$ and $\cos$ of these angles: $\sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}, \quad \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$ $\sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}, \quad \cos \left( \frac{\pi}{3} \right) = \frac{1}{2}$

Now, apply the sine addition formula: $\sin \left( \frac{7\pi}{12} \right) = \left( \frac{\sqrt{2}}{2} \times \frac{1}{2} \right) + \left( \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} \right)$ $= \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}$ $= \frac{\sqrt{2} + \sqrt{6}}{4}$

Thus, the value of the given expression is: $\sin \left( \cos^{-1} \left( \frac{\sqrt{2}}{2} \right) + \tan^{-1} \left( \sqrt{3} \right) \right) = \frac{\sqrt{2} + \sqrt{6}}{4}$

---

Would you like a detailed explanation of any of the steps, or have other questions? Here are some related questions that could be useful:

1. What are the values of $\cos^{-1} \left( \frac{1}{2} \right)$ and $\tan^{-1} (1)$?
2. How do you simplify trigonometric expressions with multiple inverse functions?
3. Can you derive the sine addition formula and apply it to other problems?
4. How do you calculate sine, cosine, and tangent for special angles like $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$?
5. What are the common trigonometric identities for solving problems involving angles?

Tip: Memorizing key values for trigonometric functions at common angles ($\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$, etc.) can save you time when solving these types of problems.