Why arctan equal to -90 degrees?

Question

Can somebody show me why $$-\arctan\left(\frac{2\pi}{1-\cos(2\pi)}\right)$$ equals to $-90^\circ$ degrees? Thanks.

Answer 1

Since $\;1-\cos 2\pi=0\;$ the argument isn't defined, but I guess the intention here is

$$\lim_{x\to 2\pi}-\arctan\frac{2\pi}{1-\cos x}=-\frac\pi2$$

since we know $\;\tan x\xrightarrow[x\to\frac\pi2\,^-]{}\infty\;$