Find \( f(x+y)f(x-y) \)
If
\[ f(x)=\frac{2^x+2^{-x}}{2} \]
then
\[ f(x+y)f(x-y) \]
is equal to
(a) \(\frac12\{f(2x)+f(2y)\}\)
(b) \(\frac12\{f(2x)-f(2y)\}\)
(c) \(\frac14\{f(2x)+f(2y)\}\)
(d) \(\frac14\{f(2x)-f(2y)\}\)
\[ f(x+y)=\frac{2^{x+y}+2^{-(x+y)}}{2} \]
\[ f(x-y)=\frac{2^{x-y}+2^{-(x-y)}}{2} \]
Multiplying,
\[ f(x+y)f(x-y) \]
\[ = \frac14 \left(2^{2x}+2^{-2x}+2^{2y}+2^{-2y}\right) \]
\[ = \frac12 \left[ \frac{2^{2x}+2^{-2x}}{2} + \frac{2^{2y}+2^{-2y}}{2} \right] \]
\[ = \frac12\{f(2x)+f(2y)\} \]
\[ \boxed{\text{Correct Answer: (a)}} \]