Why I am not getting True when testing equation?

Your expression is only true if a is positive.

Simplify[a^-x == (1/a)^x, a > 0]

True

You can use PowerExpand to find out the ratio in general:

PowerExpand[(1/a)^x a^x, Assumptions->True]

E^(2 I π x Floor[1/2 + Arg[a]/(2 π)])

For generic x this expression is only 1 when

Floor[1/2+Arg[a]/(2 π)] == 0

Using Reduce gives:

Reduce[Floor[1/2 + Arg[a]/(2 π)] == 0, a, Complexes]

(Im[a] != 0 && Re[a] < 0) || Re[a] >= 0

So the equality is only untrue for negative reals (i.e., along the branch cut for logs and powers).

Check with Simplify:

Simplify[a^-x == (1/a)^x, (Im[a] != 0 && Re[a] < 0) || Re[a] >= 0]

True