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

Problem

csc(x)−sin(x)=cos(x)*cot(x)

Solution

1. Express the left side in terms of sine and cosine using the reciprocal identity csc(x)=1/sin(x)

csc(x)−sin(x)=1/sin(x)−sin(x)

2. Find a common denominator to combine the terms into a single fraction.

1/sin(x)−sin2(x)/sin(x)=(1−sin2(x))/sin(x)

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−sin2(x)=cos2(x)

(1−sin2(x))/sin(x)=cos2(x)/sin(x)

4. Rewrite the expression as a product of two fractions to match the desired form.

cos2(x)/sin(x)=cos(x)⋅cos(x)/sin(x)

5. Substitute the quotient identity cot(x)=cos(x)/sin(x) to complete the verification.

cos(x)⋅cos(x)/sin(x)=cos(x)*cot(x)

Final Answer

csc(x)−sin(x)=cos(x)*cot(x)

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