General value of sine and cosine If
\begin{align*}
\cos(k \pi)=(-1)^k,\quad \text{and}\quad \sin(k \pi)=0,
\end{align*}then can we find generalized expressions for
\begin{align*}
\cos(\frac{k \pi}{n}),\quad k\in\mathbb{Z}, n\in \mathbb{Z^+}
\end{align*}
and
\begin{align*}
\sin(\frac{k \pi}{n}),\quad k\in\mathbb{Z}, n\in \mathbb{Z^+}.
\end{align*}
 A: This may not be the best possible answer, but a relatively comprehensive one is given in the short article Sines and Cosines for Fractions of $\pi$ by Dr. Colin Steele at The University of Manchester. 
To paraphrase the article, "$\sin\frac{\pi}{n}$ and $\cos\frac{\pi}n$ can be found for $n=2^k\times3^p\times5^q\times17^r\times257^s\times65537^t$ where $k$ is a nonnegative integer and the other indices are all $0$ or $1$." Once you have these values you can use angle sum identities to get other numerators.
For example, $\cos\frac{13\pi}{30}=\frac{1}{8}\sqrt{\sqrt{3}-\sqrt{15}+\sqrt{10+2\sqrt{5}}}$.
A: In general $$\cos n\theta= \cos^n \theta- \frac {n (n-1)}{2!} \cos^{n-2}\theta \sin^2 \theta+\cdots $$ 
$$ \sin n\theta =n cos^{n-1}\theta sin \theta-\frac {n (n-1)(n-2)}{3!} cos^{n-3}\theta sin^3\theta+\cdots$$
Here take $k=n $ and $\theta=\frac {\pi}{n} $.
A: The answer is no, you can't.
Perhaps for specific values of $n$ you could do this, like with your example
$$\cos(k\pi)=(-1)^k, \forall k \in \mathbb Z$$
In fact, you could probably do this for a lot of $n$. However, you will never find a way to do so for all $n$, because the whole purpose of the sine and cosine functions is to act as a placeholder for the infinite number of values that cannot be expressed in terms of other elementary functions. The closest thing that I can think of would be to use the fact that
$$\cos x=\sum_{i=0}^\infty \frac{(-1)^i}{(2i+1)!}x^{2i+1}$$
to say that
$$\cos \frac{k\pi}{n}=\sum_{i=0}^\infty \frac{(-1)^i}{(2i+1)!}\bigg(\frac{k\pi}{n}\bigg)^{2i+1}$$
... but that's not even closed form, so I don't think it's what you're looking for.
