Solve $\displaystyle \frac{dy}{dx} = \frac{x+ytany}{y-xtany} $
First write your ODE as
$\displaystyle (x + y \tan y)dx +(x \tan y - y)dy = 0$ then as
$\displaystyle (x \cos y + y \sin y)dx +(x \sin y - y \cos y)dy = 0$
which is of the form
$\displaystyle M(x,y)dx + N(x,y)dy = 0$. Then check $\displaystyle \dfrac{M_y-N_x}{N}$
If $\displaystyle \dfrac{M_y-N_x}{N} = Q(x)$ then $\displaystyle \mu = e^{\int Q(x)dx}$ is an integrating factor.