The solution of the differential equation \(\tan y \sec^{2} x dx+\tan x\sec^{2}y dy=0\) is

\(\tan x+\tan y=k\) \(\tan x-\tan y=k\) \(\frac{\tan x}{\tan y}=k\) \(\tan x\tan y=k\)

\(\tan x\tan y=k\)

\(\begin{aligned}\text{Given, }\tan y\sec^{x} dx&=-\tan x\sec^{2} y dy\\ \int \frac{\sec^{2} x}{\tan x}dx&=\int -\frac{\sec^{2} y}{\tan y}dy\\ \log(\tan x)&=-\log (\tan y)+\log c\\ \tan x\tan y&=k\end{aligned}\)