I am having some trouble with the following problem:

Show that the Fourier series:

$\displaystyle \frac{1}{2} a_{0} +\sum^{\infty}_{n=1} (a_{n} \cos nx +b_{n} \sin nx)$

can be written in the form

$\displaystyle \frac{1}{2} \rho_{0} +\sum^{\infty}_{n=1} \rho_{n} \cos (nx + \theta_{n})$

where $\displaystyle \rho_{n} = \sqrt{a^{2}_{n} +b^{2}_{n} }$. Express $\displaystyle \theta_{n}$ in terms of $\displaystyle a_{n}$ and $\displaystyle b_{n}$.

Ok so I got down to the fourier series being equal to this:

$\displaystyle = \frac{1}{2\pi} \int^{\pi}_{-\pi} f(\theta) d\theta + \frac{1}{\pi} \int^{\pi}_{-\pi} f(\theta) \sum^{\infty}_{n=1} \cos n(x - \theta) d\theta $

which is not exactly what we want especially the $\displaystyle \cos n(x - \theta)$.

Any suggestion??

Also any idea how to do "Express $\displaystyle \theta_{n}$ in terms of $\displaystyle a_{n}$ and $\displaystyle b_{n}$."???