Derive average rate of change formula of cos

[andrewkg]

Q.
Use the addition formula cos(u+v) = cos(u)cos(v) - sin(u)sin(v) to derive the following identity for the average rate of change of the cosine function:

(cos(x + h) - cos x) / h = cos x ((cos h - 1) / h) - sin x ((sin h) / h)

A.
cos(x+h) = cosxcosh - sinxsinh
subtitute this to (cos(x+h) - cos x)/h we get
(cos(x+h) - cos x)/h = (cosxcosh - sinxsinh -cosx)/h
then I know it must go down to =(cosx(cosh - 1) - sinxsinh)/h
but how does cosxcosh - cosx simpify to cosx(cosh-1)
?
sorry my really basic skills are poor
Thank you a ton!

 

[SteamKing]

Factor cos(x) from {[cos(x)*cos(h) - cos(x)] - sin(x)*sin(h)}

 

[haruspex]

how does cosxcosh - cosx simpify to cosx(cosh-1)

Well, what do you get if you multiply out cos(x)(cos(h)-1)?