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

Given:

$\left( \sin\left(\frac{\pi}{3}\right) \cdot \tan\left(\frac{\pi}{6}\right) + \csc\left(\frac{\pi}{4}\right) \right)^2$

Step 1: Simplify $\sin\left(\frac{\pi}{3}\right)$, $\tan\left(\frac{\pi}{6}\right)$, and $\csc\left(\frac{\pi}{4}\right)$

• $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$
• $\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$
• $\csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$

Step 2: Substitute these values into the expression

$\left( \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}} + \sqrt{2} \right)^2$

Step 3: Simplify inside the parentheses

$\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}} = \frac{1}{2}$

So the expression becomes:

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

Step 4: Expand the square

$\left( \frac{1}{2} + \sqrt{2} \right)^2 = \left(\frac{1}{2}\right)^2 + 2\left(\frac{1}{2}\right)\left(\sqrt{2}\right) + \left(\sqrt{2}\right)^2$

Step 5: Simplify the terms

$= \frac{1}{4} + 2 \cdot \frac{1}{2} \cdot \sqrt{2} + 2$

$= \frac{1}{4} + \sqrt{2} + 2$

Step 6: Combine like terms

$= 2 + \sqrt{2} + \frac{1}{4}$

Thus, the simplified result is:

$\boxed{2 + \sqrt{2} + \frac{1}{4}}$

Would you like more details or have any other questions?

Here are five related questions to explore further:

1. How do you find the exact values of trigonometric functions at special angles?
2. How do you simplify expressions involving trigonometric functions?
3. What are the properties of square roots in terms of algebraic simplifications?
4. Can you apply the same steps to more complex trigonometric expressions?
5. How do you expand and simplify binomial expressions?

Tip: Remember to memorize the key trigonometric values at special angles like $\pi/6$, $\pi/4$, and $\pi/3$; these are frequently used in calculus and trigonometry problems.