Evaluate `cos45^(@)cos30^(@)+sin45^(@)sin30^(@)`.

Answer

To evaluate the expression \( \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \), we can follow these steps:

### Step 1: Identify the values of trigonometric functions
We know the values of the trigonometric functions:
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 30^\circ = \frac{1}{2} \)

### Step 2: Substitute the values into the expression
Now, substitute these values into the expression:
\[ \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{2}\right) \]

### Step 3: Simplify each term
Calculate each term separately:
1. For \( \cos 45^\circ \cos 30^\circ \):
\[ \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2\sqrt{2}} \]

2. For \( \sin 45^\circ \sin 30^\circ \):
\[ \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{1}{2\sqrt{2}} \]

### Step 4: Combine the terms
Now, add the two results together:
\[ \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Final Result
Thus, the value of the expression \( \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \) is:
\[ \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

---