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If f(x+y)+f(x-y)=2f(x).f(y)AA x,y in R. ...

If f(x+y)+f(x-y)=2f(x).f(y)`AA x,y in R`. `f(0)ne0`, find if f(x) is even or odd function.

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f(x+y)+f(x-y)=2f(x)f(y)
put x=y=0
`2f(0)=(f(0))^2impliesf(0)=0 or f(0) = 1
given `f(0)ne0`. Hence f(0)=1
Again put y=x, and x=0
`f(x)+f(-x)=2f(0)f(x)
f(x)+f(-x)=2f(x)
`therefore` f(x)=f(-x)
Hence the function f(x) is an even function.
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