Question

what's the proof of cosy=sqrt(1-x^2)

Answer 1

\[\cos y = \sqrt{1-x^{2}}\] this is the equation

Answer 2

First some existence conditions:
The radical should exist:
\[x \in[-1,1]\]
the right-hand term should be between -1 and 1 (because the cos is the same) (you will have actually between 0 an 1, because the radical is positive)
Solving them you get the same:
\[x \in[-1,1]\]
Now just apply the inverse cos function and take into account cos periodicity, to get something like
\[y=\pm \arccos(\sqrt{1-x ^{2}})+2k \pi, k \in Z\]

Answer 3

Also take into account the definition of arccos:
arccos:[-1,1]->[0,π]