




(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;
thus
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.