Can $\tan(x)$ be expressed as a sum of $\cos$ and $\sin$?

Question

It doesn't matter if its only $\cos(x)$ and $\sin(x)$. It can $\cos(x/2)$ and $\sin(x/2)$, or even $\cos(x)$ and $\sin(x/2)$.

Answer 1

No, if you mean a sum (and not a series, i.e., an infinite sum).

Any finite sum of $\cos$'s and $\sin$'s will be bounded since $\lvert\cos\rvert \leq 1$ and $\lvert\sin\rvert \leq 1$ (namely, if you sum $k$ of them, then the sum is at most $k$). However, $\tan$ is unbounded: $$ \lim_{x\to\frac{\pi}{2}}\tan x = \infty $$

Answer 2

No, just look at the graph. $\tan x$ goes off to infinity, while $\sin x$ and $\cos x$ do not.

This is kind of a flip answer, but can be made precise. Write $\tan x=a \cos x + b \sin x$. The right cannot get larger than $|a|+|b|$ but $\tan x$ can.