Question

Calculati x:............... ​

Answer 1

$2\tan x-3\cot x-5=0 , \ x \neq \frac{k \pi}{2}, \ k \in Z \\ 2 \tan x-3 \cdot \frac{1}{\tan x}-5 =0 \\ \Rightarrow \frac{2 (\tan x)^2-3-5 \tan x}{\tan x}= 0 \\ \Rightarrow 2(\tan x)^2-3-5 \tan x=0 \\ fie \ t=\tan x , \ atunci \\ 2t^2-3-5t=0 \\ (2t+1)(t-3)= 0 \\ \Rightarrow \left \{ {2t=-1} \atop {t=3}} \right. \Longleftrightarrow \ \left \{ {t=-\frac{1}{2}} \atop {t=3}} \right. \Longleftrightarrow \left \{ {{\tan x=-\frac{1}{2}} \atop {\tan x=3}} \right.$

$\Rightarrow x=-\arctan \frac{1}{2} + k \pi, \ k \in Z \\ \Rightarrow x= \arctan 3+k \pi, \ k \in Z \\ \\ \Rightarrow \boxed{x = \left \{ {{- \arctan \frac{1}{2} +k \pi} \atop {\arctan3 +k \pi}} \right. , \ k \in Z}$