If f(x)= 2^ x +2^ -x 2 , then f(x + y)f(x - y) is equal to:

MCQ

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

✓
$\frac{1}{2}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
B
$\frac{1}{2}\big[\text{f(2}\text{x})-\text{f}(2\text{y})\big]$
C
$\frac{1}{4}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
D
$\frac{1}{4}\big[\text{f(2}\text{x})-\text{f}(2\text{y})\big]$

✓

Answer

Correct option: A.

$\frac{1}{2}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$

Given,
$\text{f(x)}=\frac{2^{\text{x}}+2^{-\text{x}}}{2}$
Now,
$\text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\Big(\frac{2^{\text{x}+\text{y}}+2^{-\text{x}-\text{y}}}{2}\Big)\Big(\frac{2^{\text{x}-\text{y}}+2^{-\text{x}+\text{y}}}{2}\Big)$
$\Rightarrow\ \text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{4}\big(2^{2\text{x}}+2^{-2\text{y}}+2^{2\text{y}}+2^{-2\text{x}}\big)$
$\Rightarrow\ \text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{2}\Big(\frac{2^{2\text{x}}+2^{-2\text{x}}}{2}+\frac{2^{2\text{y}}+2^{-2\text{y}}}{2}\Big)$
$\Rightarrow\text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{2}\big[\text{f}(2\text{x})+\text{f}(2\text{y})\big]$

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