Simplify 2cos(105)^2-1

Problem

2*cos^2(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*cos^2(θ)−1

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

2*cos^2(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*cos^2(105^∘)−1=−√(,3)/2

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