Question

Simplify the expression using trigonometric identities: sec (–θ) – cos θ.

Answer 1

Explanation:

Let's break this down. The secant of a negative angle is the same as the secant of the positive angle. This is because secant is the inverse of cosine, and cosine is an even function. f(-x) = f(x) if the function is even, and f(-x) = -f(x) if the function is odd. Sine is odd and has symmetry about the origin; cosine is even and has y-axis symmetry.

Therefore, sec(-θ) = sec(θ) and we have then

sec(θ) - cos(θ). Since secant is the inverse of cosine, we can write:

`(1)/(cos\theta)-cos\theta` and finding a common denominator:

`(1-cos^2\theta)/(cos\theta)` and using a trig identity:

`(sin^2\theta)/(cos\theta)` and simplify that down a bit by breaking it up:

`(sin\theta)/(cos\theta)*sin\theta` finally boils down to

tanθ · sinθ, the last choice there.