please any one give the following prove
a) cos -30 degree= cos 30
b) sin -30 degree = -sin 30
in which method cos terminate its sing where in case sin (-) still remain with sin
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 what definitions of 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.
This is an almost impossible question to answer because you have not told us ANYTHING about how much math you have studied. In fact, it is not even clear what question you are asking.
Are you asking why is $cos(x) = cos(-x)\ and\ sin(x) = -sin(-x)$ for all x generally?
Or are you asking why $cos(30^o) = cos(-30^o)\ and\ sin(30^o) = -sin(-30^o)$ specifically?
Do you see that answering the general question eliminates the specific question? Do you know what the unit circle is?
If you know your CAST rule:
a) Evaluating cos (-30) we find ourselves in the 4th quadrant where cos is always positive. So as a general rule: cos(-x) = cos(x)
b) Evaluation sin(-30) we find ourselves in the 4th quadrant where sin is always negative. So as a general rule: sin(-x) = -sin(x)