If the function f satisfies the relation $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)f\left(y\right)\forall x,y\in R \mathrm{and} f\left(0\right)\ne 0,$ then

• A

  f(x) is an even function

• B

  f(x) is an odd function

• C

  $\mathrm{If} f\left(2\right)=a, \mathrm{then} f\left(-2\right)=a$

• D

  $\mathrm{If} f\left(4\right)=b, \mathrm{then} f\left(-4\right)=-b$