If sin(x+y) - sin(x-y) = 2f(x)sin(y), then f(x)

From the function sin(x+y) - sin(x-y) = 2f(x)sin(y), we can deduce that f(x) = cosx

But sin(x + y) = sinxcosy + cosxsiny.

Also, sin(x - y) = sinxcosy - cosxsiny..

Hence:

sin(x+y) - sin(x-y)

= sinxcosy + cosxsiny - (sinxcosy - cosxsiny)

=  sinxcosy -  sinxcosy + cosxsiny + cosxsiny

= 2cosxsiny

2cosxsiny = 2f(x)sin(y)

Hence, f(x) = cosx

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