Let's break down the expression step by step.

The given expression is:

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

Step 1: Understand the components

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

   • The value $\tan^{-1} \left( \sqrt{3} \right)$ refers to the angle whose tangent is $\sqrt{3}$. This corresponds to an angle of $\frac{\pi}{3}$ radians (since $\tan \frac{\pi}{3} = \sqrt{3}$).

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

   • We know $\tan^{-1} \left( \sqrt{3} \right) = \frac{\pi}{3}$, so we need to add $\frac{\sqrt{2}}{2}$ to $\frac{\pi}{3}$. The sum is just a number, so we now have: $\frac{\sqrt{2}}{2} + \frac{\pi}{3}$

Step 2: Evaluate $\cos^{-1} \left( \frac{\sqrt{2}}{2} + \frac{\pi}{3} \right)$

• The expression now becomes: $\cos^{-1} \left( \frac{\sqrt{2}}{2} + \frac{\pi}{3} \right)$

  At this stage, it's important to note that $\frac{\sqrt{2}}{2}$ is a number approximately equal to 0.7071, and adding $\frac{\pi}{3}$ (about 1.047) gives us a sum that exceeds the allowable domain for $\cos^{-1}$, which only accepts values in the range $[-1, 1]$.

Thus, the expression inside the $\cos^{-1}$ function is not valid, as the sum $\frac{\sqrt{2}}{2} + \frac{\pi}{3} > 1$.

Conclusion:

The expression as written does not have a valid mathematical solution due to the invalid domain for the inverse cosine function.

Would you like help with another problem or need clarification on any step?

Related Questions:

1. How do inverse trigonometric functions work with sums inside them?
2. What are the domains of inverse trigonometric functions like $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$?
3. How do you simplify trigonometric expressions with combined inverse functions?
4. What is the range of $\cos^{-1}$ and how does it affect solutions?
5. What happens when the argument of an inverse trigonometric function is outside its domain?

Tip: Always check the range and domain of inverse functions to ensure the argument is valid before attempting to simplify or calculate values.