Question

Verify the identity: 2cosx(sin3x-sinx) = sin4x

Answer 1

$\bf \qquad \qquad \textit{triple-angle identity}\\\\ sin(3\theta)=3sin(\theta)-4sin^3(\theta)\\\\ \left. \qquad \qquad \right.\textit{quad-angle identity}\\\\ sin(4\theta)= \begin{cases} 8sin(\theta)cos^3(\theta)=4sin(\theta)cos(\theta)\\\\ \boxed{4sin(\theta)cos(\theta)-8sin^3(\theta)cos(\theta)} \end{cases}\\\\ -----------------------------\\\\ 2cos(x)[sin(3x)-sin(x)]=sin(4x)\\\\ -----------------------------$

$\bf 2cos(x)[sin(3x)-sin(x)]\implies 2cos(x)[3sin(x)-4sin^3(x)-sin(x)] \\\\\\ 2cos(x)[2sin(x)-4sin^3(x)]\implies \boxed{4sin(x)cos(x)-8sin^3(x)cos(x)}$