Beauty of phi!

We can express

$\begin{aligned} \cos \left(\dfrac{\pi}{10} \right) = \sin \left(\dfrac{2\pi}{5}\right) = \dfrac{\sqrt { a\phi + b}}{c\,(d\phi + c)} \end{aligned}$

And further express it as

$\begin{aligned}\small{\dfrac{\sqrt[p]{q}}{c}\left(\dfrac{\sqrt{ \phi\,(\phi^{m} -\psi^{m} +\phi^{n}-\psi^{n} )+\phi^{m-1} -\psi^{m-1} +\phi^{n-1} -\psi^{n-1}}}{\phi\,(\phi^{t} -\psi^{t} )+\phi^{t-1}-\psi^{t-1} }\right) }\end{aligned}$ If $S$ be the smallest possible sum of unknown positive integers excluding the sum of $t-1,m-1,n-1$ where $b$ , $c$ , $q$ , $n-1$ and $t-1$ are prime integers. Which of the following is the possible value of $S$ ?

---

Notations: $\phi = \dfrac{1+\sqrt 5}{2}$ denotes the golden ratio and $\psi = 1-\phi$ .

This is an original problem