Question

Помогите решить уравнение

tgx/2+tgx/3=0

Answer 1

sin x/2/cosx/2+sinx/3/cosx/3=[(sinx/2*cosx/3+sinx/3*cosx/2]/(c0sx/2*cosx/3)=
= sin(x/2+x/3)/(cosx/2*cosx/3)=0
x/2+x/3=5x/6=πk  
k∈Z   x=6πk/5

Answer 2

$tgx+tgy= \frac{sin(x+y)}{cosxcosy}$

$tg \frac{x}{2} +tg \frac{x}{3} =0$

$\frac{sin( \frac{x}{2} + \frac{x}{3}) }{cos \frac{x}{2}cos \frac{x}{3} } =0$

ОДЗ:
$cos \frac{x}{2} \neq 0$
$cos \frac{x}{3} \neq 0$

$\frac{x}{2} \neq \frac{ \pi }{2} + \pi k,$  k∈Z
$\frac{x}{3} \neq \frac{ \pi }{2} + \pi n,$ n∈Z

$x \neq \pi +2 \pi k,$ k∈Z
$x \neq \frac{3 \pi }{2} +3 \pi n,$ n∈Z

$sin( \frac{x}{2} + \frac{x}{3} )=0$

$sin \frac{5x}{6} =0$

$\frac{5x}{6} = \pi m,$ m∈Z

$x= \frac{6 \pi m}{5} ,$ m∈Z