Rewriting $\sin(x+\frac {\pi} {6})\cos(x)$ as $\frac {1} {4}(2\sin(2x+\frac {\pi} {6})+1)$

Question

If I have a trigonometric expression like $$\sin(x+\frac {\pi} {6})\cos(x)$$ what are the steps to simplify it to the following? $$\frac {1} {4}(2\sin(2x+\frac {\pi} {6})+1)$$

Answer 1

From $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ and $\sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)$ you get

$$\sin(a)\cos(b)=\frac12(\sin(a+b)+\sin(a-b))$$

Hence

$$\sin(x+\pi/6)\cos(x)=\frac12(\sin(2x+\pi/6)+\sin(\pi/6))$$

And since $\sin(\pi/6)=1/2$,

$$\sin(x+\pi/6)\cos(x)=\frac12(\sin(2x+\pi/6)+\frac12)=\frac14(2\sin(2x+\pi/6)+1)$$