Even, odd, or neither $y=\frac{1-\cos{x}}{1+\cos{x}}$

Question

Determine algebraically whether the given function is even, odd, or neither.

$$y=\dfrac{1-\cos{x}}{1+\cos{x}}$$ I will substitute $−x$ into the function, and then simplify. $$y(-x)=\dfrac{1-\cos(-x)}{1+\cos(-x)}=\dfrac{1-\cos{x}}{1+\cos{x}}=y(x)$$ We have $D_x:\begin{cases}x\in\mathbb{R}\\1+\cos{x}\ne0\end{cases}\Rightarrow \cos{x}\ne-1\Rightarrow x\ne\pi(2k+1),k\in\mathbb{Z}.$ We also know that the domain has to be symmetrical about $0$. Does $x\ne\pi(2k+1),k\in\mathbb{Z}$ fulfill this requirement?

Answer 1

Yes, the domain is everything except odd multiples of $\pi$, which is perfectly symmetrical, as you get $\pm$ the same odd numbers.