Challenging trig expression for solve/simplify

[asmeurer]

Take the following

>>> expr = -1 + (cos(pi*(2*n - 2)/(2*n)) + 1)*(cos(pi*(2*n - 4)/(2*(n - 1))) + 1)/((-cos(pi*(2*n - 2)/(2*n)) + 1)*(-cos(pi*(2*n - 4)/(2*(n - 1))) + 1))
>>> pprint(expr)
       ⎛   ⎛π⋅(2⋅n - 2)⎞    ⎞ ⎛   ⎛π⋅(2⋅n - 4)⎞    ⎞
       ⎜cos⎜───────────⎟ + 1⎟⋅⎜cos⎜───────────⎟ + 1⎟
       ⎝   ⎝    2⋅n    ⎠    ⎠ ⎝   ⎝  2⋅n - 2  ⎠    ⎠
-1 + ─────────────────────────────────────────────────
     ⎛     ⎛π⋅(2⋅n - 2)⎞    ⎞ ⎛     ⎛π⋅(2⋅n - 4)⎞    ⎞
     ⎜- cos⎜───────────⎟ + 1⎟⋅⎜- cos⎜───────────⎟ + 1⎟
     ⎝     ⎝    2⋅n    ⎠    ⎠ ⎝     ⎝  2⋅n - 2  ⎠    ⎠

The goal is to solve the expression for n.

The solution is GoldenRatio. simplify has a hard time simplifying this when plugged in, either in GoldenRatio form or (sqrt(5) - 1)/2 form. I was able to manually simplify it

>>> simplify(expr.subs(n, GoldenRatio).subs(1/GoldenRatio, GoldenRatio - 1)).subs(GoldenRatio - 1, 1/GoldenRatio)
0

There are probably other complex solutions as well.

I'm not able to get this from solve or solveset. solveset gets close when given the simplified version:

>>> pprint(solveset(simplify(expr)))
⎧               ⎛π⎞      ⎛ π⋅n ⎞    ⎫   ⎧               ⎛π⎞    ⎛ π⋅n ⎞      ⎛π⎞      ⎛ π⋅n ⎞        ⎫
⎨n | n ∊ ℂ ∧ cos⎜─⎟ - cos⎜─────⎟ = 0⎬ \ ⎨n | n ∊ ℂ ∧ cos⎜─⎟⋅cos⎜─────⎟ - cos⎜─⎟ + cos⎜─────⎟ - 1 = 0⎬
⎩               ⎝n⎠      ⎝n - 1⎠    ⎭   ⎩               ⎝n⎠    ⎝n - 1⎠      ⎝n⎠      ⎝n - 1⎠        ⎭

You can get the golden ratio with

>>> solveset(pi/n - pi*n/(n - 1) + 2*pi)
{1/2 + sqrt(5)/2, -sqrt(5)/2 + 1/2}

[smichr]

The following should be able to be solved for n, but I can't check it here:

simplify(expr.rewrite(tan).subs(n,(1/z+2)/2).expand()).subs(z,1/(2*n-2))

[asmeurer]

That simplifies it a great deal but solve/solevset still can't handle it.

[smichr]

can't handle it.

What are you getting? I get

>>> solve(expr)
[1/2 + sqrt(5)/2, -sqrt(5)/2 + 1/2, -sqrt(5)/2 + 3/2, sqrt(5)/2 + 3/2]

[asmeurer]

I get [], both for the original expr and the version simplified from your comment.

[jksuom]

In [1]: expr = -1 + (cos(pi*(2*n - 2)/(2*n)) + 1)*(cos(pi*(2*n - 4)/(2*(n - 1))) + 1)/((-cos(pi*(2*n - 2)/(2*n)) + 1)*(-cos(pi*(2*n - 4)/(2*(n - 1))) + 1))

In [2]: solve(expr)
Out[2]: []

In [3]: n = Symbol('n')

In [4]: expr = -1 + (cos(pi*(2*n - 2)/(2*n)) + 1)*(cos(pi*(2*n - 4)/(2*(n - 1))) + 1)/((-cos(pi*(2*n - 2)/(2*n)) + 1)*(-cos(pi*(2*n - 4)/(2*(n - 1))) + 1))

In [5]: solve(expr)
Out[5]: 
⎡1   √5    √5   1    √5   3  √5   3⎤
⎢─ + ──, - ── + ─, - ── + ─, ── + ─⎥
⎣2   2     2    2    2    2  2    2⎦

[asmeurer]

Oh crap, n is an integer by default.