  In this paper we obtain some novel identities involving trigonometric
functions. Let $n$ be any positive odd integer. We show that
$$\sum_{r=0}^{n-1}\frac1{1+\sin2\pi\frac{x+r}n+\cos2\pi\frac{x+r}n}
=\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\sin 2\pi x+\cos 2\pi x}$$ for any
complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\not\in\mathbb Z$, and
$$\sum_{j,k=0}^{n-1}\frac1{\sin 2\pi\frac{x+j}n+\sin2\pi
\frac{y+k}n}=\frac{(-1)^{(n-1)/2}n^2}{\sin 2\pi x+\sin2\pi y}$$ for all complex
numbers $x$ and $y$ with $x+y,x-y-1/2\not\in\mathbb Z$. We also determine the
values of $\prod_{k=1}^{(p-1)/2}(1+\tan\pi\frac{k^2}p)$ and
$\prod_{k=1}^{(p-1)/2}(1+\cot\pi\frac{k^2}p)$ for any odd prime $p$. In
addition, we pose several conjectures on the values of
$G_p(x)=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p})$ with $p$ an odd prime and $x$
a root of unity.
