Using the formula cos A `=sqrt((1+cos2A)/(2))`, find the value of `cos30^(@)`, it is given that `cos60^(@)=(1)/(2)`.

Answer

To find the value of \( \cos 30^\circ \) using the formula \( \cos A = \sqrt{\frac{1 + \cos 2A}{2}} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the angle**: We need to find \( \cos 30^\circ \). According to the formula, we will set \( A = 30^\circ \).

2. **Calculate \( 2A \)**: Since \( A = 30^\circ \), then \( 2A = 60^\circ \).

3. **Use the given value**: We are given that \( \cos 60^\circ = \frac{1}{2} \).

4. **Substitute into the formula**: Now we can substitute \( \cos 2A \) into the formula:
\[ \cos 30^\circ = \sqrt{\frac{1 + \cos 60^\circ}{2}} \]

5. **Substitute the value of \( \cos 60^\circ \)**:
\[ \cos 30^\circ = \sqrt{\frac{1 + \frac{1}{2}}{2}} \]

6. **Simplify the expression**:
- First, calculate \( 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \).
- Now substitute this back:
\[ \cos 30^\circ = \sqrt{\frac{\frac{3}{2}}{2}} \]

7. **Simplify the fraction**:
- Dividing \( \frac{3}{2} \) by \( 2 \) gives:
\[ \frac{3}{2} \div 2 = \frac{3}{2} \times \frac{1}{2} = \frac{3}{4} \]

8. **Take the square root**:
\[ \cos 30^\circ = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Final Answer:
Thus, the value of \( \cos 30^\circ \) is \( \frac{\sqrt{3}}{2} \).