Cauchy derivative of sin x

[William Frederick]

Hi , on page 644 of Professor Hawkins book “ God made the integers” is quoted the following derivation of the derivative of sine x
y = sine x
delta y/ delta x=
{[ sin(1/2 delta x)] /[1/2 delta x]},
X {[cos(x+ 1/2 delta x)]}
y’=cos x
by Monsieur Cauchy
Please fill in the steps for me .. I have been stumped for days
Thanks , William Frederick

 

[LCKurtz]

You need the identity [MATH]\sin a - \sin b = 2\cos(\frac{a+b} 2)\sin(\frac{a-b} 2)[/MATH].

Apply that to [MATH]\frac{\sin(x +\Delta x) - \sin(x)}{\Delta x}[/MATH]