equation solving – Solve misses positive solutions

Why Solve returns only negative solutions?

Solve((x + 1)^(2 n) + (x - 1)^(2 n) == 0, x)

$left{left{xto frac{1+e^{frac{i pi }{2 n}}}{-1+e^{frac{i pi }{2 n}}}right}right}$

But we know conjugates are also a solution i.e.

$left{left{xto -frac{1+e^{frac{i pi }{2 n}}}{-1+e^{frac{i pi }{2 n}}}right}right}$

are also solutions.

We see that

With({n = 1}, Solve((x + 1)^(2 n) + (x - 1)^(2 n) == 0, x))

returns

${{xto -i},{xto i}}$

and
$ left{left{xto frac{1+e^{frac{i pi }{2 n}}}{-1+e^{frac{i pi }{2 n}}}right}right} /. n -> 1$

returns

$i$