Question

Explain why the function y=1-cosx is an even function

Answer 1

A function is said to even if it returns the same output when given two opposite inputs:

$f(x) = f(-x)$

In your case, you have

$f(x) = 1-\cos(x), \quad f(-x) = 1-\cos(-x)$

And since $\cos(x) = \cos(-x)$, the proof is concluded.