Simplify sec(theta)-cos(theta)

Problem

sec(θ)−cos(θ)

Solution

1. Use the reciprocal identity to rewrite the secant function in terms of cosine.

sec(θ)=1/cos(θ)

2. Substitute the identity back into the original expression.

1/cos(θ)−cos(θ)

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

1/cos(θ)−cos2(θ)/cos(θ)

4. Combine the fractions over the common denominator.

(1−cos2(θ))/cos(θ)

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

sin2(θ)/cos(θ)

6. Decompose the expression to identify other trigonometric ratios.

sin(θ)⋅sin(θ)/cos(θ)

7. Use the quotient identity sin(θ)/cos(θ)=tan(θ) to reach the simplest form.

sin(θ)*tan(θ)

Final Answer

sec(θ)−cos(θ)=sin(θ)*tan(θ)

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