Value of a product of cosines and the floor of its reciprocal

Question

1. $$ \mbox{The question states}\quad {a \over b} =\prod_{n = 1 \atop{\vphantom{\LARGE A}n \not= 9}}^{17}\cos\left(n\pi \over 18\right) $$
2. $$\mbox{And it is also provided that}\quad \left\lfloor{b \over a}\right\rfloor = 85^{2} + 7^{2} + \lambda $$

The task is to find the value of $\lambda$.

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I haven't even been able to make sense of the first expression and reduce it to a sensible form. I am open to all kinds of suggestions.

Thanks in advance.

Answer 1

There is a well-known identity: $$ \prod_{n=1}^{N-1}\sin\left(\frac{\pi n}{N}\right) = \frac{2N}{2^N}\tag{1} $$ that can be proved through De Moivre's formula and/or Chebyshev polynomials.
Take $N=9$ and re-arrange to get: $$ \prod_{\substack{1\leq n\leq 17 \\ n\neq 9}}\cos\left(\frac{\pi n}{18}\right)=\frac{9}{65536}\tag{2}$$ from which $\lambda=\color{red}{7}$.