Simplify

Problem

√(,((1−cos(θ))*(1+cos(θ)))/cos^2(θ))

Solution

1. Identify the difference of squares pattern in the numerator, where (a−b)*(a+b)=a^2−b^2

2. Expand the numerator using this identity.

√(,(1−cos^2(θ))/cos^2(θ))

3. Apply the Pythagorean identity sin^2(θ)+cos^2(θ)=1 which implies 1−cos^2(θ)=sin^2(θ)

√(,sin^2(θ)/cos^2(θ))

4. Substitute the trigonometric identity tan(θ)=sin(θ)/cos(θ) into the expression.

√(,tan^2(θ))

5. Simplify the square root of the squared term, noting that √(,x^2)=|x|

|tan(θ)|

Final Answer

√(,((1−cos(θ))*(1+cos(θ)))/cos^2(θ))=|tan(θ)|

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