exponential equation real solution

edit

Well, my answer doesn't use Sage, but: your equation is equivalent to cosh(2x) == 1. The hyperbolic cosine function (cosh) has a global minimum at 0, so there's only one real solution: zero.

Other things that can help you figure this out: plot exp(2x) + exp(-2x) and see what it looks like. Show that the derivative is positive for all positive x, and negative for all negative x, and use that to prove that the function has a global minimum at 0, so that the original equation can only have one solution.

I don't see a built-in method for this, but you could solve the equation which makes the imaginary part of the general solution zero:

sage: solutions = (exp(-2*x)+exp(2*x) == 2).solve(x,to_poly_solve=True)

sage: eq1 = solutions[0]
sage: eq1
x == I*pi*z29

sage: rhs = eq1.right()
sage: rhs
I*pi*z29

sage: rhs.imag()
pi*real_part(z29)

sage: rhs.variables()
(z29,)

sage: (rhs.imag() == 0).solve(var)
[z29 == 0]

This gives the value of z29 which makes the imaginary part zero; so now substitute this value into the original equation, eq1:

sage: solutions_2 = (rhs.imag() == 0).solve(var)
sage: eq2 = solutions_2[0]

sage: eq1
x == I*pi*z29
sage: eq2
z29 == 0

sage: eq1.substitute(eq2)
x == 0

This should work for any equation you start with; you could wrap this in a function if you need to call it repeatedly (if you do that, make sure to check that there is only one solution in solutions and solutions_2, otherwise you'll want to use those equations too).