f(x-y) = f(x)f(y) – f(a-x) f(a+y) …...(1)
Replace y = -x
(1) becomes: f(2x) = f(x) f(-x) – f(a-x) f(a-x)
f(2x) = f(x) f(-x) – f^2(a-x) …..(2)
Now, put x = 0
(2) becomes: f(0) = f(0) f(0) – f^2(a)
f(0) = f^2(0) – f^2(a)
1 = 1 – f^2(a) …...(Given that f(0) = 1)
f^2(a) = 0
f(a) = 0 …..(3)
f(2a-x) = f(2a) – f(x)
= f(a+a) – f(x)
= f(a) + f(a) – f(x)
= 0 – f(x) …......(from (3))
= – f(x)
