If `cosx=(sqrt(3))/(2)`, find cos2x.

Options

A. `-0.87`
B. `-0.25`
C. `0`
D. 0.5

Explanation

To solve the problem where \( \cos x = \frac{\sqrt{3}}{2} \) and we need to find \( \cos 2x \), we can follow these steps:

### Step 1: Identify the formula for \( \cos 2x \)
The double angle formula for cosine is given by:
\[ \cos 2x = 2 \cos^2 x - 1 \]

### Step 2: Substitute the value of \( \cos x \)
We know that \( \cos x = \frac{\sqrt{3}}{2} \). We will substitute this value into the formula:
\[ \cos 2x = 2 \left( \frac{\sqrt{3}}{2} \right)^2 - 1 \]

### Step 3: Calculate \( \left( \frac{\sqrt{3}}{2} \right)^2 \)
Calculating the square:
\[ \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \]

### Step 4: Substitute back into the formula
Now substitute \( \frac{3}{4} \) back into the equation:
\[ \cos 2x = 2 \left( \frac{3}{4} \right) - 1 \]

### Step 5: Simplify the expression
Calculating \( 2 \left( \frac{3}{4} \right) \):
\[ 2 \left( \frac{3}{4} \right) = \frac{6}{4} = \frac{3}{2} \]
Now substituting this back into the equation:
\[ \cos 2x = \frac{3}{2} - 1 \]

### Step 6: Convert 1 into a fraction
To subtract, convert 1 into a fraction with the same denominator:
\[ 1 = \frac{2}{2} \]
So,
\[ \cos 2x = \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \]

### Step 7: Final answer
Thus, we find:
\[ \cos 2x = \frac{1}{2} \]

### Conclusion
The value of \( \cos 2x \) is \( \frac{1}{2} \), which corresponds to option d.