I have to show that the solution of a differential equation is an arc of a great circle. The differential equation is as follows (in spherical coordinates):

$\frac{{\mathrm{sin}}^{2}\theta {\varphi}^{\prime}}{(1+{\mathrm{sin}}^{2}\theta ({\varphi}^{\prime}{)}^{2}{)}^{\frac{1}{2}}}=C$

where $C$ is an arbitrary constant and ${\varphi}^{\prime}$ denotes the derivative of $\varphi $ with respect to $\theta $.

My reasoning:

By setting $\varphi (0)=0$, any arc of a great circle will have no change in $\varphi $ with respect to $\theta $, so with this initial condition the answer follows by proving that ${\varphi}^{\prime}=0$. My issue is that upon working this round I end up with

$({\varphi}^{\prime}{)}^{2}=\frac{{C}^{2}}{{\mathrm{sin}}^{4}\theta -{C}^{2}{\mathrm{sin}}^{2}\theta}$

From this I can see no way forward.

Where do i go from here/ what should I do instead?

$\frac{{\mathrm{sin}}^{2}\theta {\varphi}^{\prime}}{(1+{\mathrm{sin}}^{2}\theta ({\varphi}^{\prime}{)}^{2}{)}^{\frac{1}{2}}}=C$

where $C$ is an arbitrary constant and ${\varphi}^{\prime}$ denotes the derivative of $\varphi $ with respect to $\theta $.

My reasoning:

By setting $\varphi (0)=0$, any arc of a great circle will have no change in $\varphi $ with respect to $\theta $, so with this initial condition the answer follows by proving that ${\varphi}^{\prime}=0$. My issue is that upon working this round I end up with

$({\varphi}^{\prime}{)}^{2}=\frac{{C}^{2}}{{\mathrm{sin}}^{4}\theta -{C}^{2}{\mathrm{sin}}^{2}\theta}$

From this I can see no way forward.

Where do i go from here/ what should I do instead?