f(x+y)+f(x-y)=2F(x)F(y)

Solution for f(x+y)+f(x-y)=2F(x)F(y) equation:

Simplifying
f(x + y) + f(x + -1y) = 2F(x) * F(y)
(x * f + y * f) + f(x + -1y) = 2F(x) * F(y)
(fx + fy) + f(x + -1y) = 2F(x) * F(y)
fx + fy + (x * f + -1y * f) = 2F(x) * F(y)
fx + fy + (fx + -1fy) = 2F(x) * F(y)

Reorder the terms:
fx + fx + fy + -1fy = 2F(x) * F(y)

Combine like terms: fx + fx = 2fx
2fx + fy + -1fy = 2F(x) * F(y)

Combine like terms: fy + -1fy = 0
2fx + 0 = 2F(x) * F(y)
2fx = 2F(x) * F(y)

Multiply F * x
2fx = 2xF * F * y

Multiply xF * F
2fx = 2xF2 * y

Multiply xF2 * y
2fx = 2xyF2

Solving
2fx = 2xyF2

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Divide each side by '2x'.
f = yF2

Simplifying
f = yF2