To solve the given differential equation, we'll use the method of separation of variables. Let's break down the solution step by step:

1. Given differential equation:
sec^2(x) * tan(y) dx + sec^2(y) * tan(x) dy = 0

2. Rearrange the equation:
sec^2(x) * tan(y) dx = - sec^2(y) * tan(x) dy

3. Divide both sides by sec^2(x) * tan(x):
tan(y) dx = - tan(y) / tan(x) dy

4. Simplify the equation:
dx = - dy / tan(x)

5. Integrate both sides:
∫ dx = - ∫ dy / tan(x)

6. Integrate the left-hand side with respect to x:
x = - ∫ dy / tan(x)

7. Integrate the right-hand side with respect to y:
x = - ln|sec(x)| + C1, where C1 is the constant of integration

8. Rearrange the equation:
ln|sec(x)| = -x + C1

9. Take the exponential of both sides:
|sec(x)| = e^(-x+C1)

10. Remove the absolute value:
sec(x) = ±e^(-x+C1)

11. Simplify the equation:
sec(x) = ±e^C1 * e^(-x)

12. Combine the constants:
sec(x) = Ce^(-x), where C = ±e^C1

13. Take the reciprocal of both sides:
cos(x) = 1/(Ce^(-x))

14. Simplify the equation:
cos(x) = C * e^x

15. Take the reciprocal of both sides:
sec(x) = 1/(C * e^x)

16. The general solution is:
x = arccos(1/(C * e^x)) + 2nπ, where n is an integer

17. Substitute the value of x in the original differential equation:
sec^2(x) * tan(y) dx + sec^2(y) * tan(x) dy = 0

18. Simplify the equation:
sec^2(arccos(1/(C * e^x))) * tan(y) + sec^2(y) * tan(arccos(1/(C * e^x))) * dy = 0

19. Apply the trigonometric identity:
sec^2(arccos(1/(C * e^x))) = 1/(1 - (1/(C * e^x))^2) = 1/(1 - 1/(C^2 * e^(2x)))

20. Substitute the value of sec^2(arccos(1/(C * e^x))) in the equation:
1/(1 - 1/(C^2 * e^(2x))) * tan(y) + sec^2(y) * tan(arccos(1/(C * e^x))) * dy = 0

21. Rearrange the equation:
tan(y)/(1 - 1/(C^2 * e^(2x))) + sec^2(y) * tan(arccos(1/(C * e^x))) * dy = 0

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