Let f(x) = a ^ x ( a > 0 ) be written as f(x) = f _ 1 (x) + f _ 2...

$f\left(x\right) = {a}^{x}, a > 0$

$f\left(x\right) = \dfrac{{a}^{x} + {a}^{- x} + {a}^{x} - {a}^{- x}}{2}$
$⇒ {f}_{1}\left(x\right) = \dfrac{{a}^{x} + {a}^{- x}}{2}$
${f}_{2}\left(x\right) = \dfrac{{a}^{x} - {a}^{- x}}{2}$

$⇒ {f}_{1}\left(x + y\right) + {f}_{1}\left(x - y\right)$

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

$= {f}_{1}\left(x\right) \times 2{f}_{1}\left(y\right)$

$= 2{f}_{1}\left(x\right){f}_{1}\left(y\right)$