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If you track the slope of a sine wave at every point, you get a curve that peaks at zero, dips at Ο€\pi, and matches the cosine wave identically.
A visual match is a great clue, but in mathematics we need a rigorous proof from first principles using limits.
What algebraic building blocks do we need before we can evaluate this limit?

Essential Trigonometric Tools

The derivative from first principles calculates instantaneous rate of change by taking the limit of the slope between two points as their horizontal separation hh shrinks to zero:
fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
To expand trigonometric inputs like (x+h)(x+h), we use the compound angle identities:
sin⁑(x+h)=sin⁑xcos⁑h+cos⁑xsin⁑h\sin(x+h) = \sin x \cos h + \cos x \sin h cos⁑(x+h)=cos⁑xcos⁑hβˆ’sin⁑xsin⁑h\cos(x+h) = \cos x \cos h - \sin x \sin h
When the step size hh is tiny and measured in radians, trigonometry reveals two fundamental small-angle limits:
In the 1700s, Swiss mathematician Leonhard Euler championed radian measure specifically because it ensures lim⁑hβ†’0sin⁑hh=1\lim_{h \to 0} \frac{\sin h}{h} = 1 without messy conversion constants.
Now that we have our identities and limits ready, what happens when we substitute sin⁑x\sin x into the difference quotient?