Simplify (1-sin(x)^2)/(csc(x)^2-1)

Problem

(1−sin^2(x))/(csc^2(x)−1)

Solution

1. Apply Pythagorean identities to the numerator and denominator. Recall that sin^2(x)+cos^2(x)=1 which implies 1−sin^2(x)=cos^2(x) Also recall that 1+cot^2(x)=csc^2(x) which implies csc^2(x)−1=cot^2(x)

cos^2(x)/cot^2(x)

2. Rewrite the cotangent function using the quotient identity cot(x)=cos(x)/sin(x)

cos^2(x)/cos^2(x)/sin^2(x)

3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

cos^2(x)⋅sin^2(x)/cos^2(x)

4. Cancel the common factor of cos^2(x) from the numerator and the denominator.

sin^2(x)

Final Answer

(1−sin^2(x))/(csc^2(x)−1)=sin^2(x)

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