Not sure what your getting at Lee really! The identity you speak of is on of the basic identies that is used to solve many problems. I can explain its origins if that helps???? If not ignore!!!
Imagine the graph of a circle with the origin as its center and its radius of 1 unit. Now draw a Right angled triangle with the radius as its hypoyonuse which originates at the origin and obviously meets the circumference. So if we now take the angle made between the hypotonuse and the base of the triangle as X we can work out the two unknown side lengths as :
Base length: cos x = base/1
cos x = base
Opposite length: sin x = opposite/1
sin x = opposite
so now we use these values in pythagorous theorem thus...
1^2 = sin^2x + cos^2x
So the proof on the equation i suppose is just pythagorous!