Questions: Use a double-angle formula to find the exact value of the given 2 cos^2 165°-1 2 cos^2 165°-1= (Simplify your answer, including any radicals. Use integers or fractions.)

Solution Steps

The expression given is \(2 \cos^2 165^\circ - 1\). This expression resembles the double-angle identity for cosine:

\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]

We can see that the expression \(2 \cos^2 165^\circ - 1\) is equivalent to \(\cos(2 \times 165^\circ)\).

Calculate \(2 \times 165^\circ\):

\[ 2 \times 165^\circ = 330^\circ \]

The angle \(330^\circ\) is in the fourth quadrant, where the cosine is positive. The reference angle for \(330^\circ\) is \(360^\circ - 330^\circ = 30^\circ\).

Thus, \(\cos(330^\circ) = \cos(30^\circ)\).

The cosine of \(30^\circ\) is a known value:

\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

Therefore, \(\cos(330^\circ) = \frac{\sqrt{3}}{2}\).

Final Answer