Simplify 1-2sin(105)^2

Problem

1−2*sin^2(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*sin^2(θ) where θ=105^∘

1−2*sin^2(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*sin^2(105^∘)=−√(,3)/2

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