Prove an inequality involving trigonometric series

Question

I am reading a proof, in which there is an inequality that I cannot prove.

Let $\{\phi_n\}$ be a sequence of reals. Let $N$ be an integer $\ge 2$.

Let $p(t)=\sum_{n=0}^N b_n \cos(nt+\phi_n)$.

Desired inequality: $b_n \le 4/\pi \| p\|_\infty$.

I don't even know where the $4/\pi$ comes from...

Also, some of the summands can positive and some of them can be negative. How can one use the $L^\infty$ bound to derive such an inequality?

Thanks for help.

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Edit: Added the entire proof.