Simplify 2cos(165)^2-1

Problem

2*cos2(165)−1

Solution

1. Identify the expression as a form of the double-angle identity for cosine.

2. Recall the identity cos(2*θ)=2*cos2(θ)−1

3. Substitute θ=165 into the identity to rewrite the expression.

2*cos2(165)−1=cos(2⋅165)

4. Multiply the angle inside the cosine function.

cos(2⋅165)=cos(330)

5. Determine the reference angle for 330 which is in the fourth quadrant.

360−330=30

6. Evaluate the cosine of 330 using the unit circle, noting that cosine is positive in the fourth quadrant.

cos(330)=cos(30)=√(,3)/2

Final Answer

2*cos2(165)−1=√(,3)/2

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