I suppose are given as independent.

First of all you will need to derive the conditional joint CDF, then differentiate it to obtain the conditional joint pdf.

i.e. to calculate the following first:

[math] F_{U,V|U^2+V^2<1}(u,v) [/math

Now the remaining part is just a calculus problem. Sketch the region in your mind

When , the above double integral can be expressed as

and by Fundamental Theorem of Calculus,

When , we have an additional extra term

but since it is independent of , the derivative vanish.

Therefore the result follows.