If cos^2x + sin^2x = 1 could 1+cos^2x = -sin^2x? used odd even to switch sin^2(-x) to -sin^2x?
In a word , NO !!!
Both incorrect. 1 + cos^2(x) would be 2 - sin^2(x). And sin^2(-x) is not -sin^2(x), it's +sin^2(x), because "sin^2(-x)" means sin(-x)*sin(-x) = [-sin(x)]*[-sin(x)] = sin^2(x).
There's a basic arithmetic error in your calculation: cos²x + sin²x = 1 1 + ( cos²x + sin²x ) = ( 1 ) + 1 "add one on both sides to get 1 + cos²x" 1 + cos²x + sin²x = 2 1 + cos²x = 2 - sin²x
cos^2x + sin^2x = 1 1 - cos^2x = sin^2x 1 - sin^2x = cos^2x