When does Cos X = X. I want a "proof" please, I know it is at .739085133 radians but dont know "why".

Your use of quotes around "why" betrays your uncertainty as to what it is you seek. "how do I solve this equation for x ytically" is a clearer question. Not all equations can be solved ytically, and numerical methods, such as the one that gave you the answer, are required. Such is the case here.