Let's break down the expression step by step:

$\frac{\tan^2(45^\circ)}{2} + \tan(30^\circ)$

Step 1: Calculate $\tan(45^\circ)$

$\tan(45^\circ) = 1$

So, $\tan^2(45^\circ) = 1^2 = 1$.

Step 2: Calculate $\tan(30^\circ)$

$\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577$

Step 3: Substitute values into the expression

Now the expression becomes: $\frac{1}{2} + \frac{1}{\sqrt{3}}$

Step 4: Simplify

To simplify further, let's express everything with a common denominator.

$\frac{1}{2} + \frac{1}{\sqrt{3}} \approx 0.5 + 0.577 = 1.077$

The approximate value of the expression is: $\boxed{1.077}$

Let me know if you'd like more details or have questions. Here are some related problems to explore:

1. What is $\tan(60^\circ)$?
2. How do you find the exact value of $\tan(30^\circ)$?
3. What is $\frac{\tan^2(60^\circ)}{2} + \tan(45^\circ)$?
4. How does $\tan^2(45^\circ)$ relate to other trigonometric identities?
5. What is $\frac{\tan^2(30^\circ)}{2} + \tan(45^\circ)$?

Tip: When working with trigonometric functions, it’s helpful to remember the exact values for angles like 30°, 45°, and 60°.