Question

Solve : sin 2x - sin 4x + sin 6x = 0​

Answer 1

$\displaystyle\longrightarrow\sf{\sin(2x)-\sin(4x)+\sin(6x)=0}$

$\displaystyle\longrightarrow\sf{\sin(2x)+\sin(6x)=\sin(4x)}$

$\displaystyle\longrightarrow\sf{2\sin\left(\dfrac{2x+6x}{2}\right)\cos\left(\dfrac{2x-6x}{2}\right)=\sin(4x)}$

$\displaystyle\longrightarrow\sf{2\sin(4x)\cos(-2x)=\sin(4x)}$

$\displaystyle\longrightarrow\sf{2\sin(4x)\cos(2x)-\sin(4x)=0\quad\quad[\ \cos(-x)=\cos x\ ]}$

$\displaystyle\longrightarrow\sf{\sin(4x)\ \big[2\cos(2x)-1\big]=0}$

$\displaystyle\Longrightarrow\sf{\sin(4x)=0\quad;\quad\cos(2x)=\dfrac{1}{2}}$

$\displaystyle\Longrightarrow\sf{4x=n\pi\quad;\quad2x=2n\pi\pm\dfrac{\pi}{3},\quad\quad n\in\mathbb{Z}}$

$\displaystyle\longrightarrow\sf{\underline{\underline{x=\dfrac{n\pi}{4}\quad;\quad x=n\pi\pm\dfrac{\pi}{6}}}}$