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Prove difference quotient formulas for sine and cosine

by RoRi

For h \neq 0 prove

    \begin{align*} \frac{\sin (x+h) - \sin x}{h} &= \frac{\sin \left(\frac{h}{2}\right)}{h/2} \cos \left(x + \frac{h}{2} \right) \\ \phantom{=} \\ \frac{\cos (x+h) - \cos x}{h} &= - \frac{\sin \left( \frac{h}{2} \right)}{h/2} \sin \left(x + \frac{h}{2} \right). \end{align*}

Proof. We use the sine and cosine of sum and difference formulas in the following computations:

    \begin{align*} \frac{\sin (x+h) - \sin x}{h} &= \frac{ \sin \left( x + \frac{h}{2} + \frac{h}{2} \right) - \sin \left( x + \frac{h}{2} - \frac{h}{2} \right)}{h} \\ &= \frac{\sin \left(x + \frac{h}{2} \right) \cos \left( \frac{h}{2} \right) + \sin \left( \frac{h}{2} \right) \cos \left( x + \frac{h}{2} \right) - \sin \left(x + \frac{h}{2} \right) \cos \left(\frac{h}{2} \right) + \sin \left( \frac{h}{2} \right) \cos \left( x + \frac{h}{2} \right)}{h} \\ &= \frac{2 \sin \frac{h}{2} }{h} \cos \left(x + \frac{h}{2} \right) \\ &= \frac{\sin \frac{h}{2}}{h/2} \cos \left(x + \frac{h}{2} \right). \end{align*}

And,

    \begin{align*} \frac{ \cos (x+h) - \cos x}{h} &= \frac{ \cos \left( x + \frac{h}{2} + \frac{h}{2} \right) - \cos \left(x + \frac{h}{2} - \frac{h}{2} \right)}{h} \\ &= \frac{ \cos \left(x + \frac{h}{2} \right) \cos \frac{h}{2} - \sin \left( x + \frac{h}{2} \right) \sin \frac{h}{2} - \cos \left(x + \frac{h}{2} \right) \cos \frac{h}{2} - \sin \left(x + \frac{h}{2} \right) \sin \frac{h}{2}}{h}\\ &= - \frac{2 \sin \left( x + \frac{h}{2} \right) \sin \frac{h}{2}}{h} \\ &= - \frac{ \sin \frac{h}{2}}{h/2} \sin \left( x + \frac{h}{2} \right). \qquad \blacksquare \end{align*}

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