Does tactition work on both archs for the fire spray. Also this ony works on the printed arch on a turretted ship like the falcon correct?

Correct. The rear arc for the Firespray is counted as it's a primary weapon arc also. And the Falcon's only arc is the printed one to the front. Any turreted weapons usually state that they can fire outside the normal arc.

Edit: Note, if you have a secondary weapon on a Firespray, it's still restricted to the front arc only. Doesn't make any difference with Tactician, but handy to know there won't be any cannon or missile shots coming out the back.

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This.

Does tactition work on both archs for the fire spray. Also this ony works on the printed arch on a turretted ship like the falcon correct?

No, but Tactician works just as you describe!