Using the formula `cos 2theta = 2 cos^(2)theta - 1`, find the value of `cos 30^(@)`, it is being given that `cos 60^(@) = 1/2`

Answer

To find the value of \( \cos 30^\circ \) using the formula \( \cos 2\theta = 2 \cos^2 \theta - 1 \) and given that \( \cos 60^\circ = \frac{1}{2} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the angle**: We know that \( \theta = 30^\circ \). Therefore, \( 2\theta = 60^\circ \).

2. **Apply the formula**: Substitute \( \theta \) into the formula \( \cos 2\theta = 2 \cos^2 \theta - 1 \):
\[ \cos 60^\circ = 2 \cos^2 30^\circ - 1 \]

3. **Substitute the known value**: We know that \( \cos 60^\circ = \frac{1}{2} \). So we can substitute this value into the equation:
\[ \frac{1}{2} = 2 \cos^2 30^\circ - 1 \]

4. **Rearrange the equation**: Add 1 to both sides of the equation:
\[ \frac{1}{2} + 1 = 2 \cos^2 30^\circ \]
\[ \frac{3}{2} = 2 \cos^2 30^\circ \]

5. **Divide by 2**: To isolate \( \cos^2 30^\circ \), divide both sides by 2:
\[ \cos^2 30^\circ = \frac{3}{2} \div 2 = \frac{3}{4} \]

6. **Take the square root**: To find \( \cos 30^\circ \), take the square root of both sides:
\[ \cos 30^\circ = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Final Answer:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]