Need to discover what kind of mark the princess has on her armpit so you can marry her?

Simple! Just have the local friendly blacksmith cast a hollow golden calf

Hide inside the calf and have the blacksmith take it to market

The princess will obviously think this is the greatest thing ever, buy it, and set it up in her bedroom immediately

Then spy on her while she's in a state of undress to see the mark

Hold on, let me elaborate

Wait until she falls asleep, creep out of the calf, creep over to her bed, and then get a look at her armpit to see she has a mark of the sun, moon, and stars there

Which sounds less like a birthmark and more like a drunken tattoo, but whatever

Then when she wakes up, convince her to stop screaming for help long enough to fall in love over the course of a conversation