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(cosq)' = -sin and (sin)' = cos

The position of a point on the unit circle at angle q has x and y   coordinates given by p() = (cos, sin). 

As q changes slightly, the change in position is in the direction  of the tangent to the circle which is perpendicular to the position vector;


p'()=a()(-sin, cos)

By symmetry of the circle, a() is a constant independent of .

Since for very small , and sin are essentially the same, while cos is close to 1, at = 0 we have

(sin)' = cos

and so

a() = 1

which proves the claim.