Simplify

Problem

√(,((1−cos(x))*(1+cos(x)))/cos2(x))

Solution

1. Apply the difference of squares to the numerator of the fraction inside the square root.

(1−cos(x))*(1+cos(x))=1−cos2(x)

2. Use the Pythagorean identity sin2(x)+cos2(x)=1 to substitute for the numerator.

1−cos2(x)=sin2(x)

3. Rewrite the expression inside the square root using the result from the previous step.

sin2(x)/cos2(x)

4. Apply the tangent identity tan(x)=sin(x)/cos(x) to simplify the fraction.

sin2(x)/cos2(x)=tan2(x)

5. Simplify the square root of the squared trigonometric function.

√(,tan2(x))=|tan(x)|

Final Answer

√(,((1−cos(x))*(1+cos(x)))/cos2(x))=|tan(x)|

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