Math Help - quick solution required for DE of TanX*TanY=C?

1. quick solution required for DE of TanX*TanY=C?

i am very much thankful to the one who show quick solution for DE of TanX*TanY=C?

2. If I undestand correctly You require the DE the solution of which is, in implicit form,...

$\tan x\cdot \tan\ y = c$ (1)

In that case we first separe the variables...

$\tan y= c\cdot \cot x$ (2)

...then compute the logarithm of both sides of (2)...

$\ln \tan y = \ln \cot x + \ln c$ (3)

... then differenziate both sides of (3)...

$\frac{dy}{\tan y\cdot \cos^{2}y}= - \frac{dx}{\cot x\cdot \sin^{2} x}$ (4)

... and finally we obtain...

$y^{'} = -\frac {\sin y\cdot \cos y}{\sin x\cdot cos x}= -\frac{\sin 2y}{\sin 2x}$ (5)

Kind regards

$\chi$ $\sigma$