What is the value of cos `15^(@)`?

Options

A. `(1)/(2)(sqrt(2-sqrt(3)))`
B. `(1)/(2)(sqrt(2+sqrt(3)))`
C. `sqrt(2)+sqrt(3)`
D. `sqrt(2)-sqrt(3)`

Explanation

To find the value of \( \cos 15^\circ \), we can use the cosine double angle formula. Here’s the step-by-step solution:

### Step 1: Use the Cosine Double Angle Formula
The cosine double angle formula states:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
We can express \( \cos 15^\circ \) as \( \cos (30^\circ / 2) \).

### Step 2: Set Up the Equation
Let \( \theta = 15^\circ \). Then, we have:
\[ \cos 30^\circ = 2\cos^2 15^\circ - 1 \]
This can be rearranged to find \( \cos 15^\circ \):
\[ 2\cos^2 15^\circ = \cos 30^\circ + 1 \]

### Step 3: Substitute the Value of \( \cos 30^\circ \)
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Substituting this into the equation gives:
\[ 2\cos^2 15^\circ = \frac{\sqrt{3}}{2} + 1 \]

### Step 4: Simplify the Right Side
Convert 1 to a fraction with a common denominator:
\[ 1 = \frac{2}{2} \]
Thus, we have:
\[ 2\cos^2 15^\circ = \frac{\sqrt{3}}{2} + \frac{2}{2} = \frac{\sqrt{3} + 2}{2} \]

### Step 5: Solve for \( \cos^2 15^\circ \)
Now, divide both sides by 2:
\[ \cos^2 15^\circ = \frac{\sqrt{3} + 2}{4} \]

### Step 6: Take the Square Root
To find \( \cos 15^\circ \), take the square root of both sides:
\[ \cos 15^\circ = \sqrt{\frac{\sqrt{3} + 2}{4}} = \frac{\sqrt{\sqrt{3} + 2}}{2} \]

### Final Answer
Thus, the value of \( \cos 15^\circ \) is:
\[ \cos 15^\circ = \frac{\sqrt{2 + \sqrt{3}}}{2} \]