Simplify sin(x)(csc(x)-sin(x))

Problem

sin(x)*(csc(x)−sin(x))

Solution

1. Distribute the sin(x) term to both terms inside the parentheses.

sin(x)*csc(x)−sin^2(x)

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

sin(x)⋅1/sin(x)−sin^2(x)

3. Simplify the first term by canceling sin(x)

1−sin^2(x)

4. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies that 1−sin^2(x)=cos^2(x)

cos^2(x)

Final Answer

sin(x)*(csc(x)−sin(x))=cos^2(x)

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