Simplify (1-cos(x)^2)(csc(x))

Problem

(1−cos2(x))*(csc(x))

Solution

1. Identify the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−cos2(x)=sin2(x)

2. Substitute the identity into the expression to replace the first factor.

(sin2(x))*(csc(x))

3. Apply the reciprocal identity csc(x)=1/sin(x)

sin2(x)⋅1/sin(x)

4. Simplify the expression by canceling one factor of sin(x) from the numerator and denominator.

sin(x)

Final Answer

(1−cos2(x))*(csc(x))=sin(x)

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