Simplify sec(x)-1/(sec(x))

Problem

sec(x)−1/sec(x)

Solution

1. Identify the reciprocal identity for the second term, where 1/sec(x)=cos(x)

2. Substitute the identity into the expression to rewrite it in terms of secant and cosine.

sec(x)−cos(x)

3. Rewrite the secant term using the identity sec(x)=1/cos(x) to create a common denominator.

1/cos(x)−cos(x)

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

1/cos(x)−cos2(x)/cos(x)

5. Combine the fractions into a single expression.

(1−cos2(x))/cos(x)

6. Apply the Pythagorean identity 1−cos2(x)=sin2(x)

sin2(x)/cos(x)

7. Decompose the fraction to simplify into other trigonometric functions.

sin(x)⋅sin(x)/cos(x)

8. Substitute the identity sin(x)/cos(x)=tan(x)

sin(x)*tan(x)

Final Answer

sec(x)−1/sec(x)=sin(x)*tan(x)

Want more problems? Check here!