Question

решить систему tgx*tgy=1, sinx*cosy+ctgx*tgy=1/2

Answer 1

{2cosxcosy=1⇒cosxcosy=1/2
{tgx+tgy=2
sinx/cosx+siny/cosy=2
(sinxcosy+sinycosx)/coscosy=2
sin(x+y)/cosxcosy=1/2
sin(x+y)=2coscosy
sin(x+y)=2*1/2
sin(x+y)=1
x+y=π/2
y=π/2-x
tgx+tg(π/2-x)=2
tgx+ctgx=2
tgx+1/tgx-2=0
tg²x+1-2tgx=0
(tgx-1)²=0
tgx-1=0
tgx=1
x=π/4
y=π/2-π/4
y=π/4
Ответ x=π/4+πn,n∈z;y=π/4+πn,n∈z