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

Problem

sin2(x)*csc2(x)−sin2(x)

Solution

1. Identify the reciprocal identity for the cosecant function, which states that csc(x)=1/sin(x)

2. Substitute the identity into the expression to rewrite csc2(x) in terms of sine.

sin2(x)*(1/sin2(x))−sin2(x)

3. Simplify the first term by canceling the sin2(x) terms in the numerator and denominator.

1−sin2(x)

4. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which can be rearranged to 1−sin2(x)=cos2(x)

cos2(x)

Final Answer

sin2(x)*csc2(x)−sin2(x)=cos2(x)

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