The answer to this question depends upon how you define sin x and cos x.
However, an elementary approach uses a geometric figure to convince you of the 2 limits you need,
namely lim as x->0 of (sin x)/x =1 and lim x->0 of (1-cos x)/x = 0.
I can't tell you where to find the figure easily, nor the argument that goes with it (my text book is so old that it is no longer in print). I don't know how to help you any better.