Question

If $f\left(x\right)=\frac{a^x+a^{-x}}{2}$ then $f(x+y)f(x-y)$ is equal to:

Answer 1

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

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

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

$f(x+y)f(x-y)=\frac{a^x{a}^y+a^{-x}{a}^y+a^x{a}^{-y}+a^{-x}{a}^y}{4}\frac{a^x(a^y+a^{-y})+a^{-x}(a^y+a^{-y})}{4}\frac{(a^x+a^{-x})(a^y+a^{-y})}{4}f\left(x\right)f\left(y\right)$