Function even or odd?

Question

While determining fourier series expansion of following function.

f(x) = $\sqrt{1-\cos x} $

here $f(-x) = f(x)$ so it is even. But

the above function can also be written as $f(x) = \sqrt 2\sin(x/2)$

here $f(-x) = -f(x)$ so by that logic it is odd

Can anyone explain where I am wrong. Thanks in advance

Answer 1

As said @Bernard in a comment, $\forall A\in\Bbb R,\sqrt{A^2}=|A|$.

Hence $$f(x)=\sqrt{2}\sqrt{\sin^2(x/2)}=\sqrt{2}|\sin(x/2)|$$ and you can check that $f$ is even.