They've been fighting the same enemy longer than we have. No, they haven't sent troops to the theaters where we engaged them--no need to, they've got enough to deal with right there at home.
It's not that they haven't sent troops, it's that they haven't done anything at all. A couple of years ago, US and Pakistani forces were looking for bin Laden, so India decided that was a good time to send troops to the Line of Control. This forced Pakistani troops to the LOC and thus away from searching for bin Laden.
They've exporeted and sponsored terrorism as much as tehy have fought it. They also have a very loose definition of terrorism. if you peacefully try to gain sovereignty for a state or a region and you demand a free and fair vote on the matter, by the standards of the Indian government you're a terrorist and they hve the right to kill you (and just about everyone who knows you.)