In the textbook Classical Dynamics by Greenwood, in solution for geodesic problem, we come across an integral
$$\int\frac{cd\theta}{sin\theta\sqrt{sin^2\theta-c^2}}$$
where the author takes the liberty to jump straight to the answer:
$$\phi=\cos^{-1} \frac{c \cot\theta}{\sqrt{1-c^2}}+\phi_0$$
I vaguely remember doing substitutions twice to reach the answer last time I solved it, but have completely forgotten how I did it. Can someone help me?
Hint: $$\int\frac{c\ d\theta}{\sin\theta\sqrt{\sin^2\theta-c^2}}=\int\dfrac{c}{\sin^2\theta \sqrt{(1-c^2)\left(1-\frac{c^2}{1-c^2}\cot^2\theta\right)}}\ d\theta$$ now let substitution $$\cos\phi=\frac{c}{\sqrt{1-c^2}}\cot\theta$$