{sin(x+h) - sin x}/h = {2sin(h/2)cos(x+h/2)}/h

Question

Trigonometric + integration question

Answer 1

sin(a+b)=sinacosb+cosasinb; sin(a-b)=sinacosb-cosasinb.

sin(a+b)-sin(a-b)=2cosasinb.

Let a+b=x+h and a-b=x. Therefore 2a=2x+h and a=x+h/2; and 2b=h, b=h/2.

So sin(x+h)-sin(x)=2cos(x+h/2)sin(h/2).

Divide through by h and rearrange and we get {2sin(h/2)cos(x+h/2)}/h QED.