Given that $\displaystyle 2y=a^x+a^{-x}$ , where a >1 and x>0 , prove that $\displaystyle a^x=y+\sqrt{(y^2-1)}$ . Similarly , if $\displaystyle 2z=a^3x+a^{-3x}$ , prove that $\displaystyle z=4y^3-3y$
Note that $\displaystyle 2y = a^x + a^{ - x} \, \equiv \,a^{2x} - 2ya^x + 1 =0$.
If you can solve $\displaystyle z^2 -2yz +1 =0$ for z, you can do the first part.