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If tan x + tan y = [(sec x + sec y) / (cos x + cos y)] * (dy/dx), then which of the following is correct?

1. sec(x + y) - tan(x + y) = x + c
2. sec(x + y) + tan(x + y) = x + c
3. tan(x + y) - sec(x + y) = x + c
4. None of these

Correct answer: sec(x + y) + tan(x + y) = x + c

Solution

After simplification the ODE is dy/dx = sin(x+y). Substitute u = x+y, du/dx = 1 + sin u. Separate: du/(1+sin u) = dx. Multiply numerator and denominator by (1 - sin u): du*(1-sin u)/cos²(u) = dx, i.e., (sec² u - sec u tan u) du = dx. Integrate: tan u - sec u = x + c, i.e., tan(x+y) - sec(x+y) = x + c. But option (B) says sec(x+y)+tan(x+y)=x+c. The correct integral gives tan u - sec u = x+c, matching option (C).

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