Simplify

Problem

cos^2(x/2−x)/√(,1−sin^2(x))

Solution

1. Simplify the argument of the cosine function in the numerator.

x/2−x=−x/2

2. Apply the even-odd identity for cosine, cos(−θ)=cos(θ) to the numerator.

cos^2(−x/2)=cos^2(x/2)

3. Use the Pythagorean identity cos^2(x)+sin^2(x)=1 to simplify the radicand in the denominator.

1−sin^2(x)=cos^2(x)

4. Simplify the square root in the denominator, noting that √(,cos^2(x))=|cos(x)|

√(,cos^2(x))=|cos(x)|

5. Apply the power-reduction identity cos^2(θ)=(1+cos(2*θ))/2 to the numerator where θ=x/2

cos^2(x/2)=(1+cos(x))/2

6. Combine the simplified numerator and denominator into a single expression.

(1+cos(x))/(2*|cos(x)|)

Final Answer

cos^2(x/2−x)/√(,1−sin^2(x))=(1+cos(x))/(2*|cos(x)|)

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