Is this...

No. There is a prostitute but she's a ( early 70's french word for Ladyboy ).Transvestite maybe. Bloke dressed up as chick. I'm sure I've seen as least one cop-falls-for-hooker movie, just can't think of one right now...
Oh, there's Sharkey's Machine with Burt Reynolds...Actually, I think that happens quite a bit in movies...

His name's Lebowski ?! That's YOUR name Dude!

You may be thinking about Klute (1971), in which Donald Sutherland plays a small town cop who investigates a crime in New York and starts a relationship with a high-class prostitute (played by Jane Fonda). That's a rather good crime drama, and sadly unknown.

This world is a comedy to those that think, a tragedy to those that feel.

Forget "L.A Confidential" not...

Could perhaps also be thinking of Max et les ferrailleurs (1971).