derivatives of sine and cosineDerivativesOfSineAndCosine2007-04-24 19:26:192007-04-26 13:03:08DerivationProsthaphaeresis formulas% this is the default PlanetMath preamble. as your knowledge
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The \PMlinkescapetext{derivation} of the derivatives of sine and cosine is a bit simpler by using the prosthaphaeresis formulas
\begin{align}
\sin\alpha-\sin\beta = \,2\sin \left( \frac{\alpha\!-\!\beta}{2} \right) \,\cos \left( \frac{\alpha\!+\!\beta}{2} \right),
\end{align}
\begin{align}
\cos\alpha-\cos\beta = -2\sin \left( \frac{\alpha\!+\!\beta}{2} \right) \, \sin\left( \frac{\alpha\!-\!\beta}{2} \right).
\end{align}
Let $x,\,t$ be any real numbers such that\, $t \neq x$.\, Then we obtain
$$\frac{\sin{x}-\sin{t}}{x-t} =
\frac{2\sin \left( \frac{x-t}{2} \right) \cos \left( \frac{x+t}{2} \right) }{x-t} =
\frac{\sin \left( \frac{x-t}{2} \right) }{\left( \frac{x-t}{2} \right) }\cdot\cos \left( \frac{x\!+\!t}{2} \right) \;\;
\longrightarrow\; 1\cdot\cos \left( \frac{x\!+\!x}{2} \right) = \cos{x},$$
as\; $t\to x$.\, Here we used the known limit \;$\displaystyle\lim_{u\to0}\frac{\sin{u}}{u} = 1$\; (see \PMlinkname{this entry}{LimitOfDisplaystyleFracsinXxAsXApproaches0}).
The derivative of cosine is calculated similarly:
$$\frac{\cos{x}-\cos{t}}{x-t} = \frac{-2\sin \left( \frac{x+t}{2} \right) \sin\left( \frac{x-t}{2} \right)}{x-t} =-1 \cdot \frac{\sin\left( \frac{x-t}{2} \right) }{\left( \frac{x-t}{2} \right) }\cdot \sin \left( \frac{x\!+\!t}{2} \right) \;\; \longrightarrow\; -1 \cdot 1\cdot \sin \left( \frac{x\!+\!x}{2} \right) =-\sin{x},$$
as\; $t\to x$.