A) \[2{{f}_{1}}(x){{f}_{1}}(y)\]
B) \[2{{f}_{1}}(x){{f}_{2}}(y)\]
C) \[2{{f}_{1}}(x+y){{f}_{2}}(x-y)\]
D) \[2{{f}_{1}}(x+y){{f}_{1}}(x-y)\]
Correct Answer: A
Solution :
\[f(x)={{a}^{x}},a>0\] \[f(x)=\frac{{{a}^{x}}+{{a}^{-x}}+{{a}^{x}}-{{a}^{-x}}}{2}\] \[\Rightarrow \]\[{{f}_{1}}(x)=\frac{{{a}^{x}}+{{a}^{-x}}}{2}\] \[{{f}_{2}}(x)=\frac{{{a}^{x}}-{{a}^{-x}}}{2}\] \[\Rightarrow \]\[{{f}_{1}}(x+y)+{{f}_{1}}(x-y)\] \[=\frac{{{a}^{x+y}}+{{a}^{-x-y}}}{2}+\frac{{{a}^{x-y}}+{{a}^{-x+y}}}{2}\] \[=\frac{\left( {{a}^{x}}+{{a}^{-x}} \right)}{2}\left( {{a}^{y}}+{{a}^{-y}} \right)\] \[={{f}_{1}}(x)\times 2{{f}_{1}}(y)\] \[=2{{f}_{1}}(x)\times {{f}_{1}}(y)\]
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