Simplify sin(x)(1+cot(x)^2)

Problem

sin(x)*(1+cot2(x))

Solution

1. Identify the Pythagorean identity for the expression inside the parentheses.

2. Apply the identity 1+cot2(x)=csc2(x) to rewrite the expression.

sin(x)*(1+cot2(x))=sin(x)*csc2(x)

3. Use the reciprocal identity csc(x)=1/sin(x) to express the cosecant in terms of sine.

sin(x)*csc2(x)=sin(x)⋅1/sin2(x)

4. Simplify the expression by canceling one factor of sin(x) from the numerator and denominator.

sin(x)⋅1/sin2(x)=1/sin(x)

5. Rewrite the result using the reciprocal identity for cosecant.

1/sin(x)=csc(x)

Final Answer

sin(x)*(1+cot2(x))=csc(x)

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