The correct spelling would be:
omne animal triste post coitum
why triste and not tristis, you ask me. Well, the adjective is tristis, triste (a Latin second class adjective). This means that its forms are:
Masculine tristis
Feminine  tristis
Neuter    triste

And it happens that animal is a neuter noun of the third declension (animal, animalis). Since animal is neuter, the adjective that modifies it must take the neuter gender: triste

As for the source, I have seen it attributed to Ovid and Aristotle - but I can't find it in the Perseus Project.

Notice also that Latin is quite free with word order: the sentence could well be "omne animal triste post coitum" or "triste omne post coitum animal". There could as well be an est nearly anywhere.

thanks to Gritchka and mbk for various reasons