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The given expression is:

\[
\frac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ}
\]

Let’s solve this step-by-step:

### Step 1: Find the trigonometric values
- \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\)
- \(\csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2\)

### Step 2: Substitute the values into the expression
\[
\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}} + 2}
\]

### Step 3: Simplify the denominator
The denominator becomes:
\[
\frac{2}{\sqrt{3}} + 2 = \frac{2}{\sqrt{3}} + \frac{2\sqrt{3}}{\sqrt{3}} = \frac{2 + 2\sqrt{3}}{\sqrt{3}}
\]

So, the expression becomes:
\[
\frac{\frac{1}{\sqrt{2}}}{\frac{2 + 2\sqrt{3}}{\sqrt{3}}}
\]

### Step 4: Simplify the entire expression
This simplifies further as:
\[
\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2 + 2\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{2}(2 + 2\sqrt{3})}
\]

Thus, the simplified expression is:
\[
\frac{\sqrt{3}}{\sqrt{2}(2 + 2\sqrt{3})}
\]

This is the most simplified form. Do you have any questions, or would you like further explanation?

Here are 5 questions that can extend your understanding:
1. How do we rationalize the denominator for trigonometric expressions?
2. What are the exact values of all trigonometric functions for standard angles?
3. How does the reciprocal identity help in simplifying trigonometric expressions?
4. Can we approximate the final value of this expression numerically?
5. How would this expression change if a different trigonometric identity were used?

**Tip:** Always keep exact values of trigonometric functions for common angles (30°, 45°, 60°, etc.) handy; they save time when solving problems!

Evaluate \(\frac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ}\).

Learn how to simplify the trigonometric expression \(\frac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ}\). This problem involves trigonometric identities, reciprocal functions, and expression simplification. Ideal for students in Grades 9-11.

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