Simplify 2cos(105)^2-1

Problem

2*cos2(105)−1

Solution

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

2. Recall the double-angle formula for cosine, which states cos(2*θ)=2*cos2(θ)−1

3. Substitute θ=105 into the formula to rewrite the expression.

2*cos2(105)−1=cos(2⋅105)

4. Multiply the angle inside the cosine function.

cos(2⋅105)=cos(210)

5. Determine the reference angle for 210 which is 210−180=30

6. Evaluate the cosine of 210 Since 210 is in the third quadrant, the cosine value is negative.

cos(210)=−cos(30)

7. Substitute the known value of cos(30)=√(,3)/2

−cos(30)=−√(,3)/2

Final Answer

2*cos2(105)−1=−√(,3)/2

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