If f(x+y) + f(x-y) =2f(x)f(y) and f(0)=k then find f(x)?

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SUMMARY

The functional equation f(x+y) + f(x-y) = 2f(x)f(y) with the condition f(0) = k leads to the conclusion that f(x) must be of the form f(x) = cosh(ax) for some constant a. This conclusion is derived by manipulating the equation to eliminate the variable y, revealing that the function exhibits properties of hyperbolic cosine functions. The identity holds for all values of y, confirming the generality of the solution.

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if
f(x+y) + f(x-y) =2f(x)f(y)
and f(0)=k
then find f(x)?

What should be approach followed to solve this question .
please ans it soon :blushing:
 
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You have an expression with f(x) and f(y) and you only want f(x). So you have to get rid of y.

One more hint: that identity holds for all values of y.
 

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