Fourier Decomposition of cos(x)^5

The app below shows a plot of $y=cos^5\left(x\right)$ as well as $y=a\cdot cos\left(x\right)+b\cdot cos\left(3x\right)+c\cdot cos\left(5x\right)$. By playing with the sliders for try to make these two functions match.[br][br]Hint: It is useful the consider the following formula[center]﻿$cos^5\left(x\right)=\left(\frac{e^{ix}+e^{-ix}}{2}\right)^5$.[/center]