Coxeter Number Power

A zonogon is a centrally symmetric convex polygon, consisting of $2n$ sides, where $n \geq 2$ is an integer. According to the geometer Coxeter, it can be dissected into $\dfrac{1}{2}n(n - 1)$ parallelograms. Using this fact, we can establish few identities for expressing areas of a unit regular zonogon in two distinct ways:

• $\cot\dfrac{\pi}{4} = \sin\dfrac{\pi}{2}$ shows that a square can be dissected into a square (as a rhombus of interior angle $90^{\circ}$ ).
• $2\cot\dfrac{\pi}{8} = 4\sin\dfrac{\pi}{4} + 2\sin\dfrac{\pi}{2}$ shows that an octagon can be dissected into $4$ rhombi of interior angle $45^{\circ}$ and $2$ squares.
• $4\cot\dfrac{\pi}{16} = 8\sin\dfrac{\pi}{8} + 8\sin\dfrac{\pi}{4} + 8\sin\dfrac{3\pi}{8} + 4\sin\dfrac{\pi}{2}$ shows that a hexadecagon (of 16 sides) can be dissected into $8$ rhombi of interior angle $22.5^{\circ}$ , $8$ rhombi of interior angle $45^{\circ}$ , $8$ rhombi of interior angle $67.5^{\circ}$ and $4$ squares.

where these three identities are expressed in the following form

$\dfrac{n}{4}\cot\dfrac{\pi}{n} = \dfrac{n}{4}\sin\dfrac{\pi}{2} + \sum\limits_{j=1}^{\frac{n}{4} - 1} \dfrac{n}{2}\sin\dfrac{2j\pi}{n}$

How many more $n$ values in the powers of $2$ (that hold for the above identity) are there?

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Bonus: Prove or disprove that the following equation has one and only nonnegative integral $(k_1, k_2, \dots, k_{\frac{n}{4} - 1}, k_{\frac{n}{4}})$ for $n=2^m$ , where $m \geq 1$ is an integer.

$\dfrac{n}{4}\cot\dfrac{\pi}{n} = k_{\frac{n}{4}}\sin\dfrac{\pi}{2} + \sum\limits_{j=1}^{\frac{n}{4} - 1} k_j\sin\dfrac{2j\pi}{n}$

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For a harder challenge, try this problem .