Question

If \(f(x-y)+f(x+y) = 2f(x)f(y)\) for all \(x,y\in R\), then \(f(x)\) is .....

• A. an odd function
• B. an even function
• C. neither even nor odd function
• D. a periodic function

Hint:

Put x = 0, then swap x and y.

Answer 1

The Correct Option is B

Step 1: Test with cosine:
$f(x)=\cos x$ gives $\cos(x-y)+\cos(x+y)=2\cos x\cos y$. It satisfies the relation and is even.

Step 2: Rule out the others:
$f(x)=\cosh x$ also satisfies it but is not periodic, so (D) is not forced. $f(x)=\cos x$ is not odd, so (A) fails. (C) fails since examples are even.

Step 3: Prove generally:
Put $x=0$ then swap roles: $f(-y)=f(y)$ whenever $f\not\equiv0$. So even.

Final Answer:
Examples like cos x and cosh x are even, and the proof confirms it. \[ \boxed{B} \]