Complex representation of sinusoids question A sum of sinusoids defined as:
$$\tag1f(t) = \sum_{n=1}^{N}A\sin(2\pi tn) + B\cos(2\pi tn)$$ is said to be represented as:
$$\tag2f(t) = \sum_{n=-N}^{N}C\cdot e^{i2\pi tn}$$
which is derived from Euler's identity $e^{ix} = \cos(x) + i\sin(x)$ from which follows that:
$$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$ and $$\sin(x)=\frac{e^{ix} - e^{-ix}}{2i}$$ 
therefore:
$$ \cos(x) + \sin(x) = \frac{e^{ix} + e^{-ix}}{2} + \frac{e^{ix} - e^{-ix}}{2i} $$
The thing is, I cannot see how this turns into $(2)$? The last statement, no matter how I try to manipulate it, I cant get it to the form $$\cos(x) + \sin(x) = Ce^{ix}$$ so that the summation works and it makes no sense for it to be so, since Euler's equation is using $i\sin(x)$ - clearly a different thing...
Further, the statement :
$$ |\cos(x) + i\sin(x)| = 1$$ is also confusing since ${\cos(x)^2 + i\sin(x)}^2$ is not the same thing as ${\cos(x)^2 + \sin(x)}^2$ and it appears people just "drop" the $i$, which is terribly confusing.
Could someone please explain what am I not understanding? Thanks a bunch!

For a complex number $z=z+ i y$, $|z| = x^2+y^2$.  So $|e^{i t}|$ for real $t$ is $|\cos{t} + i \sin{t}| = \cos^2{t} + \sin^2{t} = 1$.
What you seek is $\cos{x} + i \sin{x} = e^{i x}$ as the complex representation.  Nobody should be dropping the $i$, ever.
