Verify the Identity cos(x)cot(x)+sin(x)=csc(x)

Problem

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

Solution

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

cos(x)⋅cos(x)/sin(x)+sin(x)

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

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

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

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

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

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

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

1/sin(x)

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

csc(x)

Final Answer

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

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