Question

Verify the following identity used in calculas: (cos(x+h) - cosx)/h = (cosxcos(h-1))/h = (sinxsinh)/h

Answer 1

definition of derivative
\[\lim _{h->0} \frac{\cos(x+h) -\cos x}{h} = \lim _{h->0} \frac{\cos x \cos h - \sin x \sin h -\cos x}{h} =? -\sin x\]
not sure how to prove it but i know the answer

Answer 2

\[= -\cos x \lim_{h->0}\frac{1-\cos h}{h} - \sin x \lim _{h->0} \frac{\sin h}{h}\]
\[= -\cos x *0 - \sin x * 1\]
\[= - \sin x\]
http://en.wikipedia.org/wiki/Trig_identities#Calculus

Answer 3

dont get it right now but i might tomorrow. thanks