To solve the expression $\frac{\cos 45^\circ}{\sec 30^\circ} + \csc 30^\circ$, we need to break it down step by step.

Step 1: Calculate 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 division

$\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2\sqrt{2}} = \frac{\sqrt{6}}{4}$

Now the expression becomes:

$\frac{\sqrt{6}}{4} + 2$

Step 4: Combine the terms

$\frac{\sqrt{6}}{4} + \frac{8}{4} = \frac{\sqrt{6} + 8}{4}$

Thus, the simplified result is:

$\boxed{\frac{\sqrt{6} + 8}{4}}$

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Would you like further details or have any questions?

Related questions:

1. How do you simplify trigonometric expressions involving multiple angles?
2. What is the geometric interpretation of trigonometric functions like secant and cosecant?
3. How do you rationalize denominators in expressions with square roots?
4. Can you explain how to derive the values for $\cos 45^\circ$, $\sec 30^\circ$, and $\csc 30^\circ$?
5. What is the relationship between trigonometric functions and their reciprocals?

Tip:

Always simplify trigonometric identities step by step and substitute known values carefully to avoid mistakes!