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

Problem

cos(x)−1/cos(x)

Solution

1. Identify the expression as a subtraction of a fraction from a trigonometric function.

2. Find a common denominator by multiplying the first term by cos(x)/cos(x)

cos2(x)/cos(x)−1/cos(x)

3. Combine the terms over the single denominator.

(cos2(x)−1)/cos(x)

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

(−sin2(x))/cos(x)

5. Rewrite the expression using the definition of the tangent function, tan(x)=sin(x)/cos(x)

−sin(x)⋅sin(x)/cos(x)

6. Simplify the final result.

−sin(x)*tan(x)

Final Answer

cos(x)−1/cos(x)=−sin(x)*tan(x)

Want more problems? Check here!