Continue nonlinsolve :
PR #11111

I found the case where complex solution is missed. When
solve_poly_system
is solving polynomial equation and that result is used in solving non polynomialequation to get other soln. May besolve_poly_system
return only real solution( so complex solution is not being used for further steps) thats why further step is with this real solution. 
It seems
solve_poly_system
is doing it’s job since it is not designed for all solution. But we havesubstitution
method usingsolveset_real
andsolveset_complex
and retuning all solutionn. In new commits I improved the code and nowsubstitution
method can solve all kind of system independently. 
I come across these kind of situation many times :
In [68]: x, y, z, r, t = symbols('x, y, z, r, t', real = True)
In [69]: soln = solveset(sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t),x,S.Reals)
In [70]: soln
Out[70]:
⎧ √r⋅│tan(t)│ ⎫
(∞, ∞) ∩ ⎪───────────────────────⎪
⎨ _____________ ⎬
⎪ ╱ 2 ⎪
⎩╲╱ tan (t) + 1 ⋅tan(t)⎭
In [71]: soln = solveset(sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t),x)
In [72]: soln
Out[72]:
⎧ √r⋅│tan(t)│ ⎫
⎪───────────────────────⎪
⎨ _____________ ⎬
⎪ ╱ 2 ⎪
⎩╲╱ tan (t) + 1 ⋅tan(t)⎭
In other words :
In [1]: Intersection(FiniteSet(x), S.Reals)
Out[1]: (∞, ∞) ∩ {x}
In [2]: Intersection(FiniteSet(x), S.Reals)
Out[2]: ℝ ∩ {x}
So I am not able to extract the x
here. Because of this nonlinsolve
returning some extra soln.
One case is in test file : test_issue_5132
.
I opened a PR for this #11280
 If someone wants to see the changes I did in
nonlinsolve
then check these gist nonlinsolve_till_24june_2016, nonlinsolve_till_21jun2016.py.
previous PRs update :

#11188  Simplified Solution for Trigonometric equation : I did some minor changes.Now I hope it is good to go.

#11234  Connecting Diophantine and solveset to get Integer solution: added some testcases. I have defined
solveset_integers
for this, it take take list of symbols(to get soln in that order).t But Currently solveset takes one symbol. I hope this PR is good to go. 
11257  solveset univariate trig inequality solvers : Actually to simplify the soln I need to use PR #11188, so if it got merged that code will help me in this PR.
Meanwhile :
continue..
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