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

Problem

csc(x)−sin(x)=cot(x)*cos(x)

Solution

1. Express the left side in terms of sine by 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 cos2(x)+sin2(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 isolate the cotangent and cosine terms.

cos2(x)/sin(x)=cos(x)/sin(x)⋅cos(x)

5. Substitute the quotient identity cot(x)=cos(x)/sin(x) to reach the right side of the identity.

cos(x)/sin(x)⋅cos(x)=cot(x)*cos(x)

Final Answer

csc(x)−sin(x)=cot(x)*cos(x)

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