Question

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

• A. $2$
• B. $4$
• C. $-2$
• D. None of these

Answer 1

The correct option is A $2$

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

On putting $y=0$, in the given relation, we $f\left(x\right)+f\left(x\right)=k\cdot f\left(x\right)\cdot f\left(0\right)$
$\Rightarrow 2f\left(x\right)=k\cdot f\left(x\right)$ .....$[\because f(0)=1]$
$\Rightarrow k=2$