How can I discretly derive $u_{n}=\sin(np)$ where $p>0$?

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How can I discretly derive $u_{n}=\sin(np)$ where $p>0$? I would show that $u_{n+1}-u_{n}\sim\ p\cos(np)$ with $u_{n}=\sin(np)$. How can I do that? I tried trigonometric formula and Taylor series but nothing worked.