Find the function $\arcsin(\cos x)$

Question is from Spivak's Calculus 15-18. It asks to find the function $\arcsin(\cos x)$. Answer below:

I have problem with the highlighted part. If that is true, then why is it our answer for $2k\pi \leq x \leq (2k+1)\pi$ is not the same as for the first case. Why do we decide to use $\cos(-x')$ instead of just $\cos(x')$?

1 Answer

$$\arcsin(\cos x)=\dfrac\pi2-\arccos(\cos x)$$

Now if $\arccos(\cos x)=y,0\le y\le\pi\ \ \ \ (1)$

and $\cos y=\cos x\implies y=2m\pi\pm x$ where $m$ is any integer such that $(1)$ is complied with.