cos(r*pi) is always irrational number except cos x = +1/2,-1/2,0,+1,-1. (r is rational number)
By proving that cos(1) is transcendental number it follows that it is irrational, check the following links:
http://math.stackexchange.com/questions/677900/proof-that-cos1-is-transcendental
http://mathworld.wolfram.com/TranscendentalNumber.html
Thanks Jamal.
But I want to prove it using pure geometric concepts or some simple trigonometric formulations. If you get any idea about it please let me know.
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