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

Problem

(1−sin2(x))/(csc2(x)−1)

Solution

1. Apply Pythagorean identities to the numerator and denominator. Recall that sin2(x)+cos2(x)=1 which implies 1−sin2(x)=cos2(x) Also recall that 1+cot2(x)=csc2(x) which implies csc2(x)−1=cot2(x)

cos2(x)/cot2(x)

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

cos2(x)/cos2(x)/sin2(x)

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

cos2(x)⋅sin2(x)/cos2(x)

4. Cancel the common factor of cos2(x) from the numerator and the denominator.

sin2(x)

Final Answer

(1−sin2(x))/(csc2(x)−1)=sin2(x)

Want more problems? Check here!