Show that under the change of variable $z=\sin ^{2} x$ the equation

$$\frac{d^{2} w}{d x^{2}}+n^{2} w=0$$

becomes

$$\frac{d^{2} w}{d z^{2}}+\frac{2 z-1}{2 z(z-1)} \frac{d w}{d z}-\frac{n^{2}}{4(z-1) z} w=0$$

Show that this is a Papperitz equation and that the corresponding $P$-function is

$$P\left\{\begin{array}{rrrr} 0 & \infty & 1 & \\ 0 & \frac{1}{2} n & 0 & z \\ \frac{1}{2} & -\frac{1}{2} n & \frac{1}{2} & \end{array}\right\}$$

Deduce that $F\left(\frac{1}{2} n,-\frac{1}{2} n ; \frac{1}{2} ; \sin ^{2} x\right)=\cos n x$.