$\sin ({\pi ^x}) = {e^x} + {e^{ - x}}$Solve for real x Friday, December 31, 2021 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.