I'm not sure what that last sentence means! I thing that "cos terminate it sing" (you meant "sign") "where in case sin(-) still remain with sin" you mean that cos(-x)= cos(x) while sin(-x)= -sin(x). But are you saying we can use that or are you saying that is what is to be proven? "cos(-x)= cos(x)" and "sin(-x)= -sin(x)" is exactly what you are asked to prove.

How you prove that depends upon exactly whatdefinitionsof sine and cosine you are using. There are several. Perhaps most common is this: if we start from the point (1, 0) on a coordinate system and measure around the circumference of the unit circle, counter-clockwise, a distance t, the ending point has coordinates (cos(t), sin(t)). If t is negative, measure a distance |t| clockwise. If that is the definition you are using, use the symmetry of the unit circle.

But you can also define cos(x) as the power series and define sin(x) as the power series . The fact that those are even and odd series, respectively, gives the proof that sin(-x)= -sin(x) and cos(-x)= cos(x).

Yet another way to define cos(x) is as the solution to the initial value problem y''= -y, y(0)= 1, y'(0)= 0 and define sin(x) as the solution to the initial value problem y''= -y, y(0)= 0, y'(0)= 1.