# tan inverse (1/x) = cot inverse (x) - pi for x less than 0 and = cot inverse x for x greater than 0. I know how to verify this but my doubt is when I'm changing functions how do I know that for like here about 0 we've different answers so how do I know that about which value I've check and will there always be only 1 such value? I mean how can we say that below 0 answer will always will same and not change I.e. cot inv x -pi . Pls show for other function like sin cos also

It depends upon the graph, here is the graph of arc tan 1/x :

Now pay attention to the part for x<0.

And now the graph of arc cotx is

Now we can see that the graph for x<0 is not same but by looking at the graph of

$-\mathrm{\pi}+{\mathrm{cot}}^{-1}\mathrm{x}\mathrm{we}\mathrm{can}\mathrm{see}\mathrm{that}\mathrm{for}\mathrm{x}0\mathrm{the}\mathrm{graph}\mathrm{is}\mathrm{same}\mathrm{so}\mathrm{that}\mathrm{is}\mathrm{why}\mathrm{they}\mathrm{are}\mathrm{equal}.\phantom{\rule{0ex}{0ex}}$

Similar approach for the arc sinx, arc cosx etc.

regards

**
**