Understanding Continuity Proof: $y = \sin x$

Question

I do not understand the following:

• I understand that taking the lim of y for delta x --> 0 is the definition of continuity of a function. Though, I am confused why the author is changing the form of the equation before taking the lim?
• Also I do not see the identities the author is using that allows for the first modification of y (2sin(deltax)/2 ... step)

Could someone help explain this?

Answer 1

1. We don't know that $\sin$ is continuous, but we know the limit $\dfrac{\sin x}{x}$ at 0. Hence the transformation to use this fact.
2. $\sin\alpha -\sin\beta=2\sin((\alpha-\beta)/2)\cos((\alpha+\beta)/2)$.

Answer 2

They use the high-school factorisation formula: $$\sin p-\sin q=2\sin\frac{p-q}2\cos\frac{p+q}2,$$ which is deduced fom the linearisation formula $$2\sin b\cos a=\sin(a+b)-\sin(a-b).$$