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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