$\sin ({\pi ^x}) = {e^x} + {e^{ - x}}$Solve for real x

Using $A.M. \ge G.M.$,

$\frac{{{e^x} + {e^{ - x}}}}{2} \ge \sqrt {{e^x}.{e^{ - x}}} $

$ \Rightarrow {e^x} + {e^{ - x}} \ge 2$

$\therefore \sin ({\pi ^x}) \ge 2$ which is not possible for any real x.

So, no solution.