Are all periodic functions either even or odd?

Answer 1

An even function is one for which f(-x) = f(x).

An odd function is one for which f(-x) = -f(x).

Cos and sec are even:

cos(-x) = cos(x)

sec(-x) = sec(x)

Sin, tan, csc, and cot are odd:

sin(-x) = -sin(x)

tan(-x) = -tan(x)

csc(-x) = -csc(x)

cot(-x) = -cot(x)

These are basic trigonometric identities.

Answer 2

No. Periodic functions will be odd or even if there is symmetry at the origin.