Question

How do you determine if `secx -cscx` is an even or odd function?

Answer 1

`sec x - csc x` is neither even nor odd.

Explanation:

• An even function is one for which `f(-x) = f(x)` for all `x` in its domain.
• An odd function is one for which `f(-x) = -f(x)` for all `x` in its domain.

Consider `f(x) = sec(x) - csc(x)` with `x=pi/4` ...

Note that:

`sec(pi/4) = 1/(cos(pi/4)) = 1/(1/(sqrt(2))) = sqrt(2)`

`csc(pi/4) = 1/(sin(pi/4)) = 1/(1/(sqrt(2))) = sqrt(2)`

`sec(-pi/4) = 1/(cos(-pi/4)) = sqrt(2)`

`csc(-pi/4) = 1/(sin(-pi/4)) = -sqrt(2)`

`f(-x) = sec(-pi/4) - csc(-pi/4) = sqrt(2)-(-sqrt(2)) = 2sqrt(2)`

`f(x) = sec(pi/4) - csc(pi/4) = sqrt(2) - sqrt(2) = 0`

So neither `f(-x) = f(x)` nor `f(-x) = -f(x)`

So `f(x)` is neither even nor odd.