Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 14E / iii / Solution

Since $1-C=1/2$, the second local hypergeometric solution isUnder $z={\sin }^{2}x$, it must be a constant multiple of the independent elementary solution $\sin (kx)$. As $x\to 0$, both $z^{1/2}$ and $x$ are asymptotic to one another, while $sin(kx)\sim kx$. Matching the normalized hypergeometric factor, whose value at zero is one, givesConsequently the hypergeometric sine identity is