Verify the Identity

Problem

sec(x)/tan(x)−tan(x)/sec(x)=cos(x)*cot(x)

Solution

1. Rewrite the terms on the left side using sine and cosine identities, where sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

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

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

3. Simplify the second fraction by multiplying by the reciprocal of the denominator.

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

4. Combine the simplified terms to form a single expression with a common denominator.

1/sin(x)−sin(x)=(1−sin2(x))/sin(x)

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

cos2(x)/sin(x)

6. Decompose the expression into a product of two trigonometric functions to match the right side of the identity.

cos(x)⋅cos(x)/sin(x)=cos(x)*cot(x)

Final Answer

sec(x)/tan(x)−tan(x)/sec(x)=cos(x)*cot(x)

Want more problems? Check here!