Question

Rewrite the product 2sin(8x) cos(5x) as a sum or difference

Answer 1

`\bf \textit{Product to Sum Identities} \\\\ sin(\alpha)sin(\beta)=\cfrac{1}{2}[cos(\alpha-\beta)\quad -\quad cos(\alpha+\beta)] \\\\\\ cos(\alpha)cos(\beta)=\cfrac{1}{2}[cos(\alpha-\beta)\quad +\quad cos(\alpha+\beta)] \\\\\\ sin(\alpha)cos(\beta)=\cfrac{1}{2}[sin(\alpha+\beta)\quad +\quad sin(\alpha-\beta)]\leftarrow \textit{let's use this one} \\\\\\ cos(\alpha)sin(\beta)=\cfrac{1}{2}[sin(\alpha+\beta)\quad -\quad sin(\alpha-\beta)] \\\\[-0.35em] \rule{34em}{0.25pt}`

`\bf 2sin(8x)cos(5x)\implies ~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~\left[ \cfrac{sin(8x+5x)+sin(8x-5x)}{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~} \right] \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill sin(13x)+sin(3x)~\hfill`

Answer 2

~THIS IS DEFINITELY INCORRECT~

This may be incorrect, I tried solving the sum.

All done on the calculator with cleared variables.

2sin(8x) = 0

cos(5x) = 1