An approximation of sin using trigonometric polynomials Consider the metric space $C_{per}[-\pi,\pi]$ (with the supremum norm and metric).
Can $sin(x)$ be approximated by a trigonometric polynomial that does not contain an appearance of $sin(x)$ itself?
Meaning, is there an approximation $p$ such that $$p\in Span \{sin(nx) | n\ne1\}\cup\{cos(nx)\}$$
for all $n\in\Bbb{N}$?
And if not, how would you show that?
 A: Idea: let be $p$ an $\epsilon$- approximation of $\sin$. The integral $\int_{-\pi}^{\pi}p\sin$ must be $\approx\int_{-\pi}^{\pi}\sin^2 = \pi$, but by the orthogonality of the trigonometrical polynomials, $\int_{-\pi}^{\pi}p\sin = 0$.
A: As pointed out in Martín-Blas Pérez Pinilla's answer, there does not exist such approximations on $[-\pi,\pi]$. However, we can prove that there is one on $[0, \pi ]$.

Consider the sequence of polynomials $(P_n)$ defined by $$P_0 = 0 \mbox{ and } \ \forall n,\ P_{n+1} = P_n + \frac{1}{2} \Big( X-P_n^2 \Big)$$
We want to prove that $(P_n)$ converges uniformly to $x \mapsto \sqrt{x}$ on $[0,1]$.
You can show, using induction, that $\forall n  \ge 0,\ \forall x \in \mathbb{R}^+,\  0 \le P_n(x) \le P_{n+1}(x) \le \sqrt{x}$.
Now we define $(\varepsilon_n)_{n \ge 0}$ by : $\varepsilon_0=1$ and $\forall n \ge 0,\ \varepsilon_{n+1} = \varepsilon_n-\frac{\varepsilon_n^2}{4}$. Note that $(\varepsilon_n) \underset{n \to +\infty}{\longrightarrow} 0$.
Firstly $\forall x \in [0,1],\ x-P_n(x)^2 \le \varepsilon_0$. Moreover, you can prove easily that $\forall n \ge 0,\ \forall x \in [0,1],\  x-P_{n+1}(x)^2 \le x-P_n(x)^2 - \frac{\big( x-P_n(x)^2 \big)^2}{4} $. It follows by induction that $$\forall n\ge 0, \forall x \in [0,1],\ x-P_n(x)^2 \le \varepsilon_n$$ 
Thus, we can conclude that $(P_n)$ converges uniformly on $[0,1]$ to $x\mapsto \sqrt{x}$.

Now we come back to your problem. Note that $\forall x \in [0,\pi],\ \sin(x) = \sqrt{1-\cos(x)^2}$.
Thus $x \mapsto P_n \big( 1 - \cos(x)^2 \big)$ converges uniformly on $[0, \pi]$ to $\sin$.
Finally, it is a well-known fact that these functions can be written as linear combination of the functions $x \mapsto \cos(px)$ for $p \ge 0$.
