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 will follow these steps:

### Step 1: Identify \( A \) and \( 2A \)
Let \( A = 30^\circ \). Then, \( 2A = 60^\circ \).

### Step 2: Use the given value of \( \cos 60^\circ \)
We know that \( \cos 60^\circ = \frac{1}{2} \).

### Step 3: Substitute into the formula
Now, substitute \( A = 30^\circ \) into the formula:
\[ \cos 30^\circ = \sqrt{\frac{1 + \cos 60^\circ}{2}} \]

### Step 4: Substitute the value of \( \cos 60^\circ \)
Replace \( \cos 60^\circ \) with \( \frac{1}{2} \):
\[ \cos 30^\circ = \sqrt{\frac{1 + \frac{1}{2}}{2}} \]

### Step 5: Simplify the expression
First, simplify the numerator:
\[ 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]
Now substitute this back:
\[ \cos 30^\circ = \sqrt{\frac{\frac{3}{2}}{2}} = \sqrt{\frac{3}{2} \cdot \frac{1}{2}} = \sqrt{\frac{3}{4}} \]

### Step 6: Final simplification
Now, simplify \( \sqrt{\frac{3}{4}} \):
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

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

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