Verify the Identity (cot(x)^2)/(csc(x))=csc(x)-sin(x)

Problem

cot2(x)/csc(x)=csc(x)−sin(x)

Solution

1. Identify the Pythagorean identity for the numerator on the left side, which states cot2(x)=csc2(x)−1

2. Substitute this identity into the expression to rewrite the numerator.

(csc2(x)−1)/csc(x)

3. Split the fraction into two separate terms by dividing each part of the numerator by the denominator.

csc2(x)/csc(x)−1/csc(x)

4. Simplify the first term by canceling one factor of csc(x) and apply the reciprocal identity sin(x)=1/csc(x) to the second term.

csc(x)−sin(x)

Final Answer

cot2(x)/csc(x)=csc(x)−sin(x)

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