Simplify 1/(cos(theta))-cos(theta)

Problem

1/cos(θ)−cos(θ)

Solution

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

1/cos(θ)−cos2(θ)/cos(θ)

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

(1−cos2(θ))/cos(θ)

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

sin2(θ)/cos(θ)

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

sin(θ)/cos(θ)⋅sin(θ)

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

tan(θ)*sin(θ)

Final Answer

1/cos(θ)−cos(θ)=tan(θ)*sin(θ)

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