Hello.
I do not think that is impossible to treat all the equations even the ones
having countable sets of solution.
A more realistic feature would be to treat cases of periodic functions like
the trigonometric ones using the following tactic.
1. The equations must be of the kind f(g(x)) = a where f is T-periodic,
and g(x) is a function.
2. Find one possible special solution b : f(b) = a.
3. Then, we would have to solve the equation g(x) = b + kT where k is an
integer and T is the period of the function f.
For example, the preceding method with cos(sqrt(x))=1/sqrt(2), we would have
:
1. b = pi / 4.
2. We solve the equation sqrt(x) = pi/4 + k*2pi assuming that k is in Z.
3. Finally, we have : x = (pi/4 + k*2pi)**2 with k in Z.
Simlarly, cos(x**2)=1/sqrt(2) would give x = sqrt(pi/4 + k*2pi) with k in Z
where we cannot more precise because this would mean that we would have to
sove pi/4 + k*2pi > 0, a problem not easy to solve in general (maybe
Semi-algebric Set Theory could manage cases of most usual polynomial
inequations).
One last example : cos(sin(x)) = 1.
1. b = 0.
2. We solve the equation sin(x) = 0 + k*2pi assuming that k is in Z.
3. Finally, ... Can sympy knows that abs(k*2pi) > 1 if k in Z \ {0} ?
Best regards.
Christophe.
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