how can i solve this equation 2cot^2(2x)+3cosec2x=0 for o<x<π??
Hello, endri!
$\displaystyle \text{Solve: }\:2\cot^2(2x)+3\csc(2x)\:=\:0\,\text{ for }0 < x < \pi.$
We have: .$\displaystyle 2\cot^2(2x) + 3\csc(2x) \:=\:0$
. . . . $\displaystyle 2[\csc^2(2x) - 1] + 3\csc(2x) \:=\:0 $
. . . . . $\displaystyle 2\csc^2(2x) - 2 + 3\csc(2x) \:=\:0$
. . . . . $\displaystyle 2\csc^2(2x) + 3\csc(2x) - 2 \:=\:0$
. . . .$\displaystyle [2\csc(2x) - 1][\csc(2x)+2] \:=\:0$
Then: .$\displaystyle \begin{Bmatrix}2\csc(2x) - 1 \:=\:0 & \Rightarrow & \csc(2x) \:=\:\frac{1}{2} && \text{No real roots} \\ \csc(2x) + 2\:=\:0 & \Rightarrow & \csc(2x) \:=\:-2 & \Rightarrow & 2x \:=\:\frac{7\pi}{6},\,\frac{11\pi}{6} & \Rightarrow & \boxed{x \:=\:\tfrac{7\pi}{12},\,\tfrac{11\pi}{12}} \end{Bmatrix}$