I'm thinking about the application(s) of the quotient $\frac{\sin (\theta)}{\sin (\frac{\theta}{p})}$ and $\frac{\cos (\theta)}{\cos(\frac{\theta}{p})}$, where $p=2,3,4...$ I can only think of the formula $$ U_n(\cos\theta)=\frac{\sin ((n+1)\theta)}{\sin\theta},\quad n=0,1,2,3..., $$ where $U_n(x)$ is the Chebyshev polynomials of the second kind. (http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html)
Any suggestion, idea, or comment is welcome, thanks!