This problem may be difficult, as stated. Let me show the best possible inequality $\max|A_1|<4/\pi$, where the maximum is taken over ALL trigonometric sums that you wrote (so that $n$ is not fixed.) Consider the polynomial $$P(z)=\sum_{1}^n A_nz^n;$$ This is a polynomial with real coefficients and your restriction is equivalent to saying that $|\mathrm{Re}\, P(z)|\leq 1$ for $|z|\leq 1$. Let us maximize $|A_1|$ in the class of ALL analytic functions in the unit disk, with real coefficients.
The extremal function is the conformal map of the unit disk onto the vertical strip $|\mathrm{Re}\, x|<1$. This is easily proved using Schwarz Lemma. The explicit form of this extremal function is $$f(z)=\frac{2i}{\pi}{\mathrm{Log}}\frac{1-iz}{1+iz}=\frac{4}{\pi}\left(z-z^3/3+z^5/5-z^7/7+\ldots\right).$$ This proves the inequality stated above. It is best possible, since this $f$ can be approximated by polynomials (of high degree).