Question

How do you prove `Sin(x/3)cos(x/3) = (1/2)sin(2x/3)`?

Answer 1

We substitute `theta=x/3` in the `sin(2 theta ) = 2 sin theta cos theta`
and we're home.

Explanation:

The double angle formula for sine is

`sin(2 theta ) = 2 sin theta cos theta`

That's an old saw your teacher will generally accept without proof.

Setting `theta = x/3`,

`sin(2 cdot x/3) = 2 sin (x/3) cos(x/3)`

`1 /2 sin((2x}/3) = sin (x/3) cos(x/3) quad sqrt`