Simplify sin(x)csc(-x)

Problem

sin(x)*csc(−x)

Solution

1. Apply the odd-even identity for the cosecant function, which states that csc(−x)=−csc(x)

sin(x)*csc(−x)=sin(x)*(−csc(x))

2. Use the reciprocal identity for cosecant, where csc(x)=1/sin(x)

sin(x)*(−csc(x))=sin(x)*(−1/sin(x))

3. Multiply and simplify the expression by canceling the sin(x) terms.

sin(x)*(−1/sin(x))=−sin(x)/sin(x)

4. Reduce the fraction to its simplest form.

−sin(x)/sin(x)=−1

Final Answer

sin(x)*csc(−x)=−1

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