Evaluate: lim h --> 0 (cos(x+h) - cos x / h) Describe the result of this limit.?

The answer is as follows:

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I think "lim h --> 0 (cos(x+h) - cos x / h)" means
lim h→0 (cos(x+h) - cos(x))/h.

We know
cos(A+B) - cos(A-B)
= (cosAcosB - sinAsinB) - (cosAcosB + sinAsinB)
= -2sinAsinB

So if A+B=P and A-B=Q then A=(P+Q)/2 and B=(P-Q)/2, and
cosP - cosQ = -2sin[(P+Q)/2]*sin[(P-Q)/2].

Therefore
lim h→0 (cos(x+h) - cos(x))/h
= lim h→0 -2sin[(2x+h)/2]*sin[h/2]/h
= lim h→0 -sin[x+h/2]*sin[h/2]/(h/2)
= -sin(x)