The expression provided in the image is:

$A = \frac{3 \tan(60^\circ) - \sin(45^\circ) + \tan(60^\circ) \cdot \tan(30^\circ)}{1 + \tan(60^\circ) \cdot \tan(30^\circ)}$

Let's solve this step-by-step:

1. Trigonometric Values:

   • $\tan(60^\circ) = \sqrt{3}$
   • $\sin(45^\circ) = \frac{\sqrt{2}}{2}$
   • $\tan(30^\circ) = \frac{1}{\sqrt{3}}$

2. Substitute these values into the equation:

$A = \frac{3 \cdot \sqrt{3} - \frac{\sqrt{2}}{2} + \sqrt{3} \cdot \frac{1}{\sqrt{3}}}{1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}}}$

3. Simplify the expression:

   • In the numerator: $\sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1$, so the numerator becomes: $3 \sqrt{3} - \frac{\sqrt{2}}{2} + 1$
   • In the denominator: $\sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1$, so the denominator becomes: $1 + 1 = 2$

4. Final expression:

$A = \frac{3 \sqrt{3} - \frac{\sqrt{2}}{2} + 1}{2}$

This is the simplified form of the expression. If you need further numerical simplification or have any questions, feel free to ask!

Would you like a detailed explanation of any of the steps? Here are 5 related questions you might find helpful:

1. How do you simplify square roots involving trigonometric values?
2. Can you explain how to use trigonometric identities to solve more complex problems?
3. What are the exact values of basic trigonometric functions for common angles like 30°, 45°, and 60°?
4. How do you handle expressions involving both sine and tangent functions?
5. What are some applications of solving such trigonometric equations?

Tip: Always ensure your angle units (degrees or radians) match when solving trigonometric equations!