The given differential equation is

⇒ tanysec²xdx + tanxsec²ydy = 0

Divide throughout by tanxtany

Integrate

Put tanx = t hence that is sec²xdx = dt

Put tany = z hence that is sec²ydy = dt

⇒ log t + log z + c = 0

Resubstitute t and z

⇒ log(tan x) + log(tan y) + c = 0

Using log a + log b = log ab

⇒ log(tan x tan y) = -c

⇒ tan x tan y = e^-c

e is a constant -c is a constant hence e^-c is a constant, let it be denoted as k hence k = e^-c

⇒ tan x tan y = k