Show that this set of functions are linearly independent Consider the set $\{\cos{x},\sin{x},\cos{2x}, \sin{2x},\cdots\}\cup\{1\}\cup\{e^{\cos x}\}$.
I was able to show that the first two sets are linearly independent (there are many solutions out there as well) but I am stuck in showing the independence of the last set. I tried to assume that they are linearly dependent such that
$$0=a_0+ \sum_{n=1}^N a_n\cos nx+b_n\sin nx+ c_n e^{\cos x} $$ for some coefficients $a_i,b_i,c_i$. Can anyone give me some hints on this? Thanks!
 A: A complex analytic proof:  From the formulas for $\cos (2x), \cos (3x)$ etc we can write $e^{\cos x}=p(\cos x)$ for some polynomial $p$. It follows that the entire functions $e^{z}$ and $p$ coincide on a set with a limit point. By the Identity Theorem we see that $e^{z}=p(z)$ for all $z$ which is a contradiction (For example,  LHS has no zeros).
Second proof: As mentioned above it is a well known fact that $\cos (nx) $ can be expressed as a polynomial in $\cos x$. Hence we get $e^{\cos x} =\sum_{i=0}^{m} b_i \cos^{i}x$ for some $m$ and some coefficients $b_i$. This gives $e^{t}= \sum_{i=0}^{m} b_i t^{i}$ for $-1 \leq t \leq 1$.  Now differentiate $m+1$ times w.r.t. $t$ and put $t=0$. You get $1=0$, a contradiction.
A: The Fourier representation of a function is unique. It would suffice to show that all the Fourier coefficients are non-zero.
Indeed, the function $x\to e^{\cos x}$ is positive, even and decreasing on $[0,\pi]$. When we integrate such a function against $\cos kx$ to get the $k$-th coefficient, we get a positive result since the main contribution to the integral (on $[0,\pi]$) arises from the interval $[0,\frac{\pi}{2k}]$ which is the closest to zero. The negative contribution from $[\frac{\pi}{2k},\frac{\pi}{k} ]$  is smaller in absolute value, and the same happens with every pair of subsequent contiguous intervals, giving a positive outcome.
By uniqueness, you cannot represent the given function as a finite linear combination of the Fourier basis.
This argument actually shows that any function that is positive, decreasing on $[0,\pi]$ and even will require all the cosines of multiple arcs in their representation.
