Simplify cos(x)*cot(x)

Problem

cos(x)*cot(x)

Solution

1. Apply the identity for the cotangent function, which is defined as the ratio of the cosine to the sine.

cot(x)=cos(x)/sin(x)

2. Substitute this identity into the original expression.

cos(x)⋅cos(x)/sin(x)

3. Multiply the terms in the numerator.

cos2(x)/sin(x)

4. Apply the Pythagorean identity cos2(x)=1−sin2(x) to rewrite the numerator.

(1−sin2(x))/sin(x)

5. Split the fraction into two separate terms.

1/sin(x)−sin2(x)/sin(x)

6. Simplify each term using the reciprocal identity csc(x)=1/sin(x) and basic division.

csc(x)−sin(x)

Final Answer

cos(x)*cot(x)=csc(x)−sin(x)

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