Find difference quotient for $f(x) = \sin x$

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I need to use the different quotient:

$\frac{f(x+h)-f(x)}{h}$

to show that

$f(x) = \sin(x)$

simplifies to

$\cos(x) \frac{\sin(h)}{h} + \sin(x) \frac{\cos(h)-1}{h}$

How do I do that? I already have, working from the start point,

$\frac{\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}{h}$

but I don't know where to go from there. Thank you!

EDIT:

Following a hint from the first comment, I now have

$\frac{(\sin(x))(\cos(h) - 1)}{h} + \frac{\cos(x)\sin(h)}{h}$

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1 Answer

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Things are much easier if we choose to use Prosthaphaeresis Formula,

$$\sin(x+h)-\sin x=2\sin\dfrac h2\cos\dfrac{2x+h}2$$

and use $\lim_{u\to0}\dfrac{\sin u}u=1$

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