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. $\sqrt{3}$

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Final Answer: $\frac{\sqrt{3}}{2}$

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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: $\cos 2 θ = 2 \cos^{2} θ - 1$

Step 2 ：We can rearrange this to match the given expression: $2 \cos^{2} θ - 1 = \cos 2 θ$

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: $\frac{\sqrt{3}}{2}$

Use a double-angle identity to find the exact value of the expression.2cos2⁡165∘−1Choose the correct answer below.A. 12B. −3c. 32D. −32E. −12F. 3

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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. $\sqrt{3}$