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Draw a unit circle and pick a point, say in the first quadrant at angle x. Pick another point at an angle x+x’ and draw a line connecting the points. As x’->0, the line will become tangent to the circle. The length of the line will approach x’ (assuming x, x’ measured in radians). The change in sin(x) will be the vertical component of the line, which, using basic geometry, one can show will be the length of the line multiplied by the cosine of the angle x. Thus, in the limit of small x’, sin(x+x’)-sin(x)= x’cos(x), so d/dx(sin(x))=cos(x)
