Simplify (csc(x)-sin(x))/(cos(x))

Problem

(csc(x)−sin(x))/cos(x)

Solution

1. Substitute the reciprocal identity csc(x)=1/sin(x) into the numerator.

(1/sin(x)−sin(x))/cos(x)

2. Find a common denominator for the terms in the numerator by multiplying sin(x) by sin(x)/sin(x)

(1/sin(x)−sin2(x)/sin(x))/cos(x)

3. Combine the fractions in the numerator.

(1−sin2(x))/sin(x)/cos(x)

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

cos2(x)/sin(x)/cos(x)

5. Simplify the complex fraction by dividing the numerator by cos(x)

cos2(x)/(sin(x)*cos(x))

6. Cancel the common factor of cos(x) from the numerator and denominator.

cos(x)/sin(x)

7. Apply the quotient identity cos(x)/sin(x)=cot(x)

cot(x)

Final Answer

(csc(x)−sin(x))/cos(x)=cot(x)

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