1. The problem states the integral equation: $$\int \frac{\sin x}{\cos x} \, dx + \int \frac{\sin y}{\cos y} \, dy = \int 0$$. 2. Recognize that the integrand $$\frac{\sin x}{\cos x}$$ can be rewritten using a trigonometric identity: $$\frac{\sin x}{\cos x} = \tan x$$. 3. So the integrals become $$\int \tan x \, dx + \int \tan y \, dy = \int 0$$. 4. The integral of $$\tan x$$ is $$-\ln|\cos x| + C$$, so integrating with respect to both variables we get: $$-\ln|\cos x| + (-\ln|\cos y|) = C'$$. 5. Rearranging, this is: $$-\ln|\cos x| - \ln|\cos y| = -\ln|C|$$. 6. The method used here is **integration by substitution**, treating $$u=\cos x$$ and similarly for $$y$$, where the derivative of cosine appears as a factor inside the integrand structure allowing straightforward integration. 7. The constant $$C$$ is combined from both integrals. Final answer: The integration method used is substitution to handle $$\tan$$ integrals.