Use a half-angle identity to find the exact value. \[ \cos 15^{\circ} \]

Step 1 ：Given the problem, we need to find the exact value of \( \cos 15^{\circ} \).

Step 2 ：We can use the half-angle identity for cosine, which is given by: \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \).

Step 3 ：To find the exact value of \( \cos 15^{\circ} \), we set \( \frac{\theta}{2} = 15^{\circ} \), which gives \( \theta = 30^{\circ} \).

Step 4 ：The cosine of 30 degrees is a known value, \( \cos 30^{\circ} = 0.8660254037844387 \).

Step 5 ：We substitute this value into the identity to find the exact value of \( \cos 15^{\circ} \), which gives \( \cos 15^{\circ} = \sqrt{\frac{1 + 0.8660254037844387}{2}} = 0.9659258262890683 \).

Step 6 ：Final Answer: The exact value of \( \cos 15^{\circ} \) is \( \boxed{0.9659258262890683} \).