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I unfortunately don't have time to check why this doesn't work, but the following keyword argument (useful to know in any case) does work:

sage: (exp(-2*x)+exp(2*x) == 2).solve(x,to_poly_solve=True)
[x == I*pi*z15]

which should be interpreted as saying z15 is a free integer variable (hence z). One can also do

sage: (exp(-x)+exp(x) == 2).solve(x,to_poly_solve='force')
[x == 2*I*pi*z24]

for all possible solutions to the original one. Unfortunately, finding the documentation for this (currently) requires using the method notation instead of functional

sage: solve?