Simplify 1-2sin(105)^2

Problem

1−2*sin2(105)

Solution

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

2. Apply the formula cos(2*θ)=1−2*sin2(θ) where θ=105

1−2*sin2(105)=cos(2⋅105)

3. Multiply the angle inside the cosine function.

cos(210)

4. Determine the reference angle and quadrant for 210 Since 210 is in the third quadrant, the cosine value is negative.

Reference Angle=210−180=30

5. Evaluate the cosine of the reference angle.

cos(210)=−cos(30)

cos(30)=√(,3)/2

6. Substitute the value back into the expression.

−√(,3)/2

Final Answer

1−2*sin2(105)=−√(,3)/2

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