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

Problem

sin^2(x)*csc^2(x)−sin^2(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 csc^2(x) in terms of sine.

sin^2(x)*(1/sin^2(x))−sin^2(x)

3. Simplify the first term by canceling the sin^2(x) terms in the numerator and denominator.

1−sin^2(x)

4. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which can be rearranged to 1−sin^2(x)=cos^2(x)

cos^2(x)

Final Answer

sin^2(x)*csc^2(x)−sin^2(x)=cos^2(x)

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