Verify the Identity (cos(x))/(sec(x)sin(x))=csc(x)-sin(x)

Problem

cos(x)/(sec(x)*sin(x))=csc(x)−sin(x)

Solution

1. Rewrite the left side using the reciprocal identity for secant, where sec(x)=1/cos(x)

cos(x)/(1/cos(x)*sin(x))

2. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

cos2(x)/sin(x)

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

(1−sin2(x))/sin(x)

4. Split the fraction into two separate terms.

1/sin(x)−sin2(x)/sin(x)

5. Simplify each term using the reciprocal identity csc(x)=1/sin(x) and reducing the second fraction.

csc(x)−sin(x)

Final Answer

cos(x)/(sec(x)*sin(x))=csc(x)−sin(x)

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