Question

Solve the following equations.
$\sin x+\sin 2x=\tan x.$

Answer 1

$\sin x(1+2\cos x)=\frac{\sin x}{\cos x}$

Let $\cos x=z.$

$z+{2z}^2-1=0$

${2z}^2+2z-z-1=0$

$(z+1)(2z-1)=0$

$\therefore x=-1.$ and $2z-1=0⟹z=\frac{1}{2}.$

So, $x=(2n+1)\pi$ and $x=2n\pi \pm \frac{\pi }{3}.$

Also, $\sin x=0⟹x=n\pi .$