Question

How do you determine if `tanx +secx` is an even or odd function?

Answer 1

`tan(x)+sec(x)` is neither even nor odd.

Explanation:

• `f(x)` is an even function if it satisfies `f(-x) = f(x)` for all `x` in its domain.
• `f(x)` is an odd function if it satisfies `f(-x) = -f(x)` for all `x` in its domain.

We find:

• `tan(-x) = -tan(x)` so `tan(x)` is a non-zero odd function.
• `sec(-x) = 1/cos(-x) = 1/cos(x) = sec(x)` so `sec(x)` is a non-zero even function.

Hence `tan(x) + sec(x)` is neither odd nor even, being the sum of a non-zero odd function and a non-zero even function.

For example:

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

`tan(pi/4)+sec(pi/4) = 1+sqrt(2)`

So:

`tan(-pi/4)+sec(-pi/4) != tan(pi/4)+sec(pi/4)`

`tan(-pi/4)+sec(-pi/4) != -(tan(pi/4)+sec(pi/4))`