Question

Solve $1+2cosecx=-\frac{se{c}^2\left(x/2\right)}{2}$.

Answer 1

$1+2\cdot \frac{1+t^2}{2t}=-\frac{1}{2}(1+t^2)$ where $t=\tan \frac{x}{2}$
$t^3+2t^2+3t+2=0$ or $(t+1)(t^2+t+2)=0$
$\therefore t=-1$ only as other factor gives imaginary values
$\therefore \frac{x}{2}=n\pi +(-\frac{\pi }{4})$
$\therefore x=2n\pi -\frac{\pi }{2}$.