How to solve the equation

2^(sin(x)^4-cos(x)^2)-2^(cos(x)^4-sin(x)^2) = cos(2*x)

symbolically? The solve command produces a weird answer. Evalfing all its values, one sees

0.7853981634, -0.7853981634, 2.356194490, -2.356194490,

1.570796327 - 1.031718534 I, -1.570796327 + 1.031718534 I,

1.570796327 + 1.031718534 I, -1.570796327 - 1.031718534 I,

0.7853981634, -0.7853981634, 2.356194490, -2.356194490,

1.570796327 - 1.031718534 I, -1.570796327 + 1.031718534 I,

1.570796327 + 1.031718534 I, -1.570796327 - 1.031718534 I

The identify command

interprets the real solutions on -Pi..Pi as -3*Pi/4, -Pi/4, Pi/4, 3*Pi/4

(for example,

identify(2.356194490);

3*Pi/4 ).

Is it possible to obtain these with Maple in a simpler way?

PS. Mathematica 10 does the job.

PPS. So does even Mathematica 7.