A faucet that is meant to have a sidespray has an integral diverter machined into the assembly that diverts the water to the sprayer from the faucet arm when the trigger is pressed, but the water to the faucet must be turned on which also allows you to achieve the desired temperature. If you just tap into a water line for the sprayer, it will always be pressurized. This is recipe for serious water damage when the light-duty sprayer or hose fails, probably when everyone is away or asleep. There are commercial grade sprayers that are designed for this type of use, but that isn't what you have and even they should be turned off when not in use.

Ok. So it really can't be done. Thanks for the explanation. I appreciate it.

Consider the kohler hirise side spray. There a few threads on gardenweb about it. It is a little pricey but would work since your current faucet does not accept a side spray.
Good Luck!

Here is a link that might be useful: Kohler Hirise