Calculate $\arctan x$ if $\arcsin x$ is known? I haven't been able to figure out using conventional identities, to calculate another if one is known. Say $\arcsin x$ is known, what is $\arctan x$ in terms of the former term?
This is not as easy as it looks.
How do I use the laws of trigonometry to solve this deceptively simple problem?

This looks like a bit of a cheat!!  $tan^{-1}(x)$=$tan^{-1}(sin(sin^{-1}x))$
The only other thing that comes to mind is something similar to $sin^{-1}(x)+cos^{-1}(x)=\pi/2$
A quick look on Wolfram Alpha suggests many formulae but none explicitly use $sin^{-1}(x)$ so I suppose there is no answer to your question. But it's an interesting question. 
