CALLED the forward difference operatorWe KNOW, Δf(x) = f(x + h) – f(x)Where, h = interval of differencing = 1 (given)∴ Δsin (4X) = sin (4 (x + 1)) – sin 4x∵ \(sinC - sinD = 2\cos \left( {\frac{{C + D}}{2}} \right).\sin \left( {\frac{{C - D}}{2}} \right)\)= \(2\cos \left( {\frac{{4\left( {x + 1} \right) + 4x}}{2}} \right).\sin \left( {\frac{{4\left( {x + 1} \right) - 4x}}{2}} \right)\)= 2 cos (2 (2X + 1)) . sin 2