Let's calculate the values of \( \cos 45^\circ \) and \( \tan 30^\circ \).

### Step 1: Value of \( \cos 45^\circ \)
From trigonometric tables or unit circle knowledge:
\[
\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

### Step 2: Value of \( \tan 30^\circ \)
\[
\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

### Step 3: Add the two values
\[
\cos 45^\circ + \tan 30^\circ = \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{3}
\]

### Step 4: Find the common denominator and simplify
The common denominator of 2 and 3 is 6:
\[
\frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{6}, \quad \frac{\sqrt{3}}{3} = \frac{2\sqrt{3}}{6}
\]
Now, adding the two fractions:
\[
\frac{3\sqrt{2}}{6} + \frac{2\sqrt{3}}{6} = \frac{3\sqrt{2} + 2\sqrt{3}}{6}
\]

So, the result is:
\[
\cos 45^\circ + \tan 30^\circ = \frac{3\sqrt{2} + 2\sqrt{3}}{6}
\]

Would you like more details or have any questions?

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Here are 5 related questions you might consider:
1. What is the value of \( \sin 45^\circ + \cos 30^\circ \)?
2. How do you find exact trigonometric values using the unit circle?
3. What are the trigonometric values of special angles like 30°, 45°, and 60°?
4. Can you derive the value of \( \tan 60^\circ \) using a 30°-60°-90° triangle?
5. What is the value of \( \sin(45^\circ) \cdot \tan(30^\circ) \)?

**Tip**: Familiarize yourself with trigonometric values of common angles (30°, 45°, 60°) as they are frequently used in problems.

Learn how to solve the trigonometric expression cos 45° + tan 30° with step-by-step calculations. Understand the values of special angles and how to add trigonometric fractions. Suitable for students in Grades 9-12.

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