Simplify sin(x)+cos(x)*cot(x)

Problem

sin(x)+cos(x)*cot(x)

Solution

1. Rewrite the cotangent function using the quotient identity cot(x)=cos(x)/sin(x)

sin(x)+cos(x)cos(x)/sin(x)

2. Multiply the terms in the second part of the expression.

sin(x)+cos2(x)/sin(x)

3. Find a common denominator by multiplying the first term by sin(x)/sin(x)

sin2(x)/sin(x)+cos2(x)/sin(x)

4. Combine the fractions over the single denominator sin(x)

(sin2(x)+cos2(x))/sin(x)

5. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the numerator.

1/sin(x)

6. Use the reciprocal identity 1/sin(x)=csc(x) to reach the final simplified form.

csc(x)

Final Answer

sin(x)+cos(x)*cot(x)=csc(x)

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