derivatives of sine and cosine

The of the derivatives of sine and cosine is a bit simpler by using the prosthaphaeresis formulas

$\sin \alpha -\sin \beta =\mathrm{\hspace{0.17em}2}\sin \left({\displaystyle \frac{\alpha -\beta }{2}}\right)\cos \left({\displaystyle \frac{\alpha +\beta }{2}}\right),$ (1)

$\cos \alpha -\cos \beta =-2\sin \left({\displaystyle \frac{\alpha +\beta }{2}}\right)\sin \left({\displaystyle \frac{\alpha -\beta }{2}}\right).$ (2)

Let $x,t$ be any real numbers such that  $t\ne x$.  Then we obtain

$$\frac{\sin x-\sin t}{x-t}=\frac{2\sin \left(\frac{x-t}{2}\right)\cos \left(\frac{x+t}{2}\right)}{x-t}=\frac{\sin \left(\frac{x-t}{2}\right)}{\left(\frac{x-t}{2}\right)}\cdot \cos \left(\frac{x+t}{2}\right)⟶\mathrm{\hspace{0.33em}1}\cdot \cos \left(\frac{x+x}{2}\right)=\cos x,$$

as  $t\to x$.  Here we used the known limit  $\underset{u\to 0}{lim}{\displaystyle \frac{\sin u}{u}}=1$  (see this entry (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0)).

The derivative of cosine is calculated similarly:

$$\frac{\cos x-\cos t}{x-t}=\frac{-2\sin \left(\frac{x+t}{2}\right)\sin \left(\frac{x-t}{2}\right)}{x-t}=-1\cdot \frac{\sin \left(\frac{x-t}{2}\right)}{\left(\frac{x-t}{2}\right)}\cdot \sin \left(\frac{x+t}{2}\right)⟶-1\cdot 1\cdot \sin \left(\frac{x+x}{2}\right)=-\sin x,$$

as  $t\to x$.

Title	derivatives of sine and cosine
Canonical name	DerivativesOfSineAndCosine
Date of creation	2013-03-22 16:58:58
Last modified on	2013-03-22 16:58:58
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Derivation
Classification	msc 26A09
Related topic	DerivativesOfSinXAndCosX
Related topic	LimitOfDisplaystyleFracsinXxAsXApproaches0
Related topic	DefinitionsInTrigonometry
Related topic	LimitRulesOfFunctions