\[ y''(x)=-\frac {1}{4} y(x) \csc ^2(x) \left (-4 n^2+4 v (v+1) \sin ^2(x)-\cos ^2(x)+2\right ) \] ✓ Mathematica : cpu = 0.58564 (sec), leaf count = 42
\[\left \{\left \{y(x)\to c_1 \sqrt [4]{\cos ^2(x)-1} P_v^n(\cos (x))+c_2 \sqrt [4]{\cos ^2(x)-1} Q_v^n(\cos (x))\right \}\right \}\] ✓ Maple : cpu = 0.299 (sec), leaf count = 113
\[\left \{y \left (x \right ) = \frac {\sqrt {-2 \cos \left (2 x \right )+2}\, \left (2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \left (c_{1} \hypergeom \left (\left [\frac {n}{2}-\frac {v}{2}, \frac {n}{2}+\frac {v}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+c_{2} \sqrt {2 \cos \left (2 x \right )+2}\, \hypergeom \left (\left [\frac {n}{2}+\frac {v}{2}+1, \frac {n}{2}-\frac {v}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right ) \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}}}{\sqrt {\sin \left (2 x \right )}}\right \}\]