Use a double-angle identity to find the exact value of the expression. \[ 2 \cos ^{2} 165^{\circ}-1 \] Choose the correct answer below. A. $\frac{1}{2}$ B. $-\sqrt{3}$ c. $\frac{\sqrt{3}}{2}$ D. $-\frac{\sqrt{3}}{2}$ E. $-\frac{1}{2}$ F. $\sqrt{3}$

Step 1 ：The given expression is in the form of a double angle identity for cosine. The double angle identity for cosine is given by: \(\cos 2\theta = 2\cos^2\theta - 1\)

Step 2 ：We can rearrange this to match the given expression: \(2\cos^2\theta - 1 = \cos 2\theta\)

Step 3 ：So, we need to find the value of \(\cos(2*165°)\).

Step 4 ：The value of \(\cos(2*165°)\) is approximately 0.866. However, we need to find the exact value.

Step 5 ：The closest value to 0.866 among the options is \(\frac{\sqrt{3}}{2}\), which is approximately 0.866. Therefore, the exact value of the expression is \(\frac{\sqrt{3}}{2}\).

Step 6 ：Final Answer: \(\boxed{\frac{\sqrt{3}}{2}}\)