Military Policy & Weapons/Aircraft carrier
Quick question , I am intrigued as to why does American navy planes land on aircraft carriers using a hook & line method which is risky - when most of these planes have the vertical landing and take off facility which is lot safer.
Well, I think you're under a mistaken impression here. You say that "most of these planes have the vertical landing and takeoff facility..."
This isn't the case. There is currently only one American fighter aircraft with vertical takeoff and landing capability, and it's the Marine Corps' [not Navy] AV-8B Harrier II. The Harrier doesn't operate from the typical US Navy aircraft carriers either; they fly from Marine assault carriers, which also carry AH-1 Cobra attack helicopters and various other transport helos.
Now, the F-35B, the Marine Corps' version of the Joint Strike Fighter, will have vertical takeoff and landing capability, but it isn't in service yet. And when the JSF does enter full production, the Navy's version [the F-35C] won't be VTOL either, but rather the conventional catapult launch/arrestor wire recovery method that you mentioned at the beginning of your question. The British Royal Navy will also field the VTOL F-35B [which is only fair, you might say, since the Harrier was a British design back then].
No question, having VTOL [Vertical Take Off and Landing] brings with it some serious advantages. Not only are carrier landings easier [and less dangerous], but you don't need a base of any kind, floating or land, to set down on. Any clearing larger than the aircraft will do in a pinch. This is why the Marines love the Harrier so much, because it can be forward-deployed with the grunts it's designed to support.
VTOL also has disadvantages, though. Two words: payload and range. Taking off vertically requires a LOT of thrust, and the more the jet is carrying, the more thrust is required to get into the air. This means the maximum payload [weapons] is considerably less than that of a conventional-takeoff aircraft. Furthermore, a VTOL engine system takes up more space within the aircraft's fuselage. On the Harrier, there are four separate exhaust nozzles attached to the engine, which rotate to direct thrust depending on the direction of flight. This extra space required for the more complex engine means you have less room to carry fuel, which therefore means you have less range. This limitation is even more apparent in the F-35B, which has a complete second turbofan devoted to vertical lift. True, with aerial refueling, you can extend an aircraft's range, but bringing a tanker into a combat area isn't the best idea, now, is it? No, tankers orbit a good distance away from any danger, so combat range is measured from that distance.
So, VTOL aircraft definitely have their advantages, and there will always be a need for that capability. But they're not going to replace conventional aircraft, which have much longer effective combat range, and can carry a bigger weapons load.
One more thing I want to mention, about your statement that VTOL is a "lot safer." Remember that Naval and Marine aviators who fly off conventional aircraft carriers and land by catching arrestor wires, are extremely, extremely well trained in this. They do this literally hundreds of times in training before being deployed with the fleet. Is it completely safe? No, of course not. But the high level of training makes it safer than you might think.
Thanks for the great question. I hope I've been helpful.