Tan (x) + tan (y) = 2

2*cos (x) * cos (y) = 1

решить систему уравнений

{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