I consider myself a decent sniper, but when I go into Crucible, I'm wrecked beyond belief. There are guys who headshot me the very second they scope in. It's -blam!-ing ridiculous! I don't understand how they do it!
EDIT: A lot of people are saying that they get their skill from Halo. However, I actually snipe very well in Halo, but not-so-well in Destiny.
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Don't try and line the shot up before you fire if someone pops out away from where you're ads. Sweep across and fire. Usually the quickest reaction time wins, not necessarily who has the shot perfectly lined up (although it does help).