I stumbled upon this equation: $$\exp_a\left\{\sum_{k=1}^n \gcd(k,n)\cos\left(\frac{2\pi k}{n}\right)\right\}\equiv 1\pmod n\tag*{$\big(\forall a\in\mathbb{Z}\land n\in\mathbb{Z}^+\big)$}$$ such that $\exp_a\left\{b\right\} = a^b$ for all $a,b\in\mathbb{R}$ and $e=2.7182818\ldots$ is the natural base. Is this equation true at all? How can it be proven? I tried looking at similar equations, and doing some research and some exploration through memory, the following theorem looks fairly similar:
Fermat's Little Theorem:
$$a^{p-1}\equiv 1\pmod p.\tag*{$\big(\forall a\in\mathbb{Z}$ and prime $p\big)$}$$
However, the modulus is a prime $p$ in this theorem, but in the upmost equation, the modulus is a positive integer $n$. Could I turn this around somehow, or am I not looking in the right direction?
I found this odd equation in images by looking at math equations to set as my desktop background, and I am now sharing this to you in just different notation (as it looks prettier in my opinion). I cannot find the actual image unfortunately, but it and the equation above is nearly identical.
Nota Bene (Please Note): My skill level is not that great when solving/proving such equations...
Thank you in advance.
Because $\; \phi(n) = \sum_{k=1} ^n \gcd(k, n) \cos(2 \pi k / n)\; $ and $\; a^{\phi(n)} \equiv 1 \pmod n \;$ your equation follows. The congruence is the natural generalization of Fermat's little theorem. The sum equation for Euler's totient function is mentioned in the Wikipedia article and in MSE question 2757228 "Derivation of the gcd formula for the nth cyclotomic polynomial".