Given the function f ( x ) = a x + a - x 2 , ( a > 2 ) then f ( x + y ) + f ( x - y ) is equal to

Given that, $f\left(x\right)=\frac{a^x+a^{-x}}{2}, \left(a>2\right) ....\left(\mathrm{i}\right)$

$\Rightarrow f\left(x+y\right)=\frac{a^{x+y}+a^{-\left(x+y\right)}}{2}$ and $f\left(x-y\right)=\frac{a^{x-y}+a^{-(x-y)}}{2}$

Now, $f(x+y)+f(x-y)=\frac{a^{x+y}+a^{-x-y}}{2}+\frac{a^{x-y}+a^{-x+y}}{2}$

$=a^y\left(\frac{a^x+a^{-x}}{2}\right)+a^{-y}\left(\frac{a^x+a^{-x}}{2}\right)$

$=a^y·f\left(x\right)+a^{-y}·f\left(x\right)$ [from $\left(\mathrm{i}\right)$]

$=2·f\left(x\right)·\left(\frac{a^y+a^{-y}}{2}\right)$

$=2·f\left(x\right)·f\left(y\right)$