Question

Is the function `f(x) = 2 cot x` even, odd or neither?

Answer 1

`2cot(x)` is an odd function.

Explanation:

A function `f(x)` is even if and only if `f(-x) = f(x)`
A function `f(x)` is odd if and only if `f(-x) = -f(x)`

Note that `sin(x)` is an odd function and `cos(x)` is even.

Thus we have
`f(-x) = 2cot(-x) = 2cos(-x)/sin(-x)`

Because sine is an odd function and cosine is even, we have

`f(-x)= 2cos(x)/(-sin(x)) = -(2cos(x)/sin(x)) = -f(x)`

Thus `2cot(x)` is odd.