Find the general solution of the differential equations `sec^2xtany dx+sec^2ytanx dy=0`

Options

A. `tan y tan x=c`
B. `(tan y)/(tan x)=c`
C. `(tan^(2)x)/(tan y)=c`
D. None of these

Answer

A

Explanation

Given `(sec^(2))/(tanx)dx=-(sec^(2)y)/(tany)dy`
`impliesint(sec^(2)x)/(tanx)dx=-int(sec^(2)y)/(tany)dy`
Put `tanx=u`
`impliessec^(2)x dx=du`
and `tany=v`
`implies sec^(2)y dy =dv`
`:.int(du)/u=-int(dv)/v`

`implieslogu=-logv+logc`
`impliesuv=c`
`:.tanxtany=c`