Find all integers $m$ and $n$ such that $n\cos\frac{\pi}{m}= \sqrt{8+\sqrt{32+\sqrt{768}}}$ 
Find all integers $m$ and $n$ such that: 
  $$n\cos\frac{\pi}{m}= \sqrt{8+\sqrt{32+\sqrt{768}}}$$

I tried to find the relation of $m$ and $n$, but it's difficult for me. I need help! Thanks! 
 A: $$8+\sqrt{32+\sqrt{768}}=8+\sqrt{8(\sqrt3+1)^2}=8\left(1+\dfrac{\sqrt3+1}{2\sqrt2}\right)=8(1+\cos(60-45)^\circ)=16\cos^2(7.5^\circ)$$
A: By applying the formula
$$\cos x=\sqrt{\frac{1+\cos(2x)}2}$$
three times, you can obtain $n\cos k=$

By direct comparison, $$n= 4$$
$$\cos 8k=\frac12\implies 8k=\pm \frac\pi3$$
Thus, $m=\pm24$.
To prove uniqueness, consider
$$|m|\ge2\implies\frac1{|m|}\le\frac12\implies \frac\pi{|m|}\le\frac\pi2$$
In the region, cosine is positive. Thus, $n>0$.
For a moment, assume $m>0$.
Since $n\cos x$ is strictly decreasing in the region $0\le x\le\pi/2$, it intersects a single value(a horizontal line) at most once. Thus $m=24, n=4$ is the unique solution for $m,n>0$.
Exploiting the evenness of cosine, $m=-24, n=4$ is also a solution.
ADDED:
One may question whether, with $a,b,c,d$
natural numbers and $b,d$ non-square numbers, the following is true:
$$\sqrt{a+\sqrt{b}}=\sqrt{c+\sqrt{d}}\implies a=c, b=d$$
Suppose $a\ne c, b\ne d$, $a,b,c,d$ are natural numbers, $b,d$ non square numbers.
Manipulating $\sqrt{a+\sqrt{b}}=\sqrt{c+\sqrt{d}}$ by repeatedly squaring both sides and rearranging to get rid of square roots, one obtains
$$((a-c)^2-(b-d))^2=4(a-c)^2d$$
Note that the left hand side is a square number, thus so as the right hand side. Then, this requires either $d$ to be a square number, or $(a-c)=0$, which both oppose our supposition.
Thus, $$\sqrt{a+\sqrt{b}}=\sqrt{c+\sqrt{d}}\implies a=c, b=d$$ is true.
Similarly, the direct comparison I made is therefore justified.
p.s. Personally, I like @labbhattacharjee ‘s answer more than mine. His answer doesn’t require the ADDED part proof. Well, I think of half angle formula because of the square roots; his answer needs some math sense to see the hidden $\cos(60-45)$.
