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

Problem

cot(x)/sec(x)=csc(x)−sin(x)

Solution

1. Rewrite the left side of the equation using the fundamental trigonometric identities cot(x)=cos(x)/sin(x) and sec(x)=1/cos(x)

cot(x)/sec(x)=cos(x)/sin(x)/1/cos(x)

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

cos(x)/sin(x)⋅cos(x)=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 by dividing each part of the numerator by the denominator.

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

cot(x)/sec(x)=csc(x)−sin(x)

Want more problems? Check here!