Question

Solve the following equation:
$\cot x-2\sin 2x=1.$

Answer 1

$\cot x-2\sin 2x=1$

$\cos x-{4\sin }^{2}x\cos x=1$

$\cos x-4(1-{cos}^2)\cos x=1$

Let $\cos x=z.$

$z-4(z-z^3)=1$

${4z}^3+3z+1=0$

$(x+1){(2x-1)}^2=0$

So, $\cos x=-1$ and $\cos x=\frac{1}{2}.$

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