1st order Differential Equation

Is it possible to solve this differential equation? I can't seem to separate the variables.

$\frac{d\theta }{dv}=\frac{\frac{-cos\theta }{v}+v}{-sin\theta }$

Thanks

Yes it is solvable. It's a nonlinear ODE but you can bring it to a Bernoulli equation form:

$\dfrac{\mathrm{d} v}{\mathrm{d} \theta} + f(\theta) v + g(\theta) v^{n} = 0$

for some functions $f,g$ I leave for you to determine after correct rearrangement, and $n \neq 0,1$.

The Bernoulli substitution $v = u^{\frac{1}{1-n}}$ will bring this ODE down to a form you can solve with standard methods because it will make the ODE a linear one.