Simplify 1/(cos(x))-cos(x)

Problem

1/cos(x)−cos(x)

Solution

1. Find a common denominator by multiplying the second term by cos(x)/cos(x)

1/cos(x)−cos2(x)/cos(x)

2. Combine the fractions into a single expression over the common denominator.

(1−cos2(x))/cos(x)

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−cos2(x)=sin2(x)

sin2(x)/cos(x)

4. Rewrite the expression as a product of two fractions to identify standard trigonometric functions.

sin(x)/cos(x)⋅sin(x)

5. Substitute the identity tan(x)=sin(x)/cos(x) to reach the simplest form.

tan(x)*sin(x)

Final Answer

1/cos(x)−cos(x)=tan(x)*sin(x)

Want more problems? Check here!