Question : Simplify the following: $\sin2x+2\sin4x+\sin6x$
Option 1: $4\cos^2x\sin4x$
Option 2: $4\cos^2x\sin x$
Option 3: $2\cos^2x\sin4 x$
Option 4: $4\sin^2 x\sin4 x$
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Correct Answer: $4\cos^2x\sin4x$
Solution : Given: $\sin2x+2\sin4x+\sin6x$ $=\sin2x+\sin6x+2\sin4x$ Using the formula: $\sin C+\sin D=2\sin\frac{(C+D)}{2}\cos\frac{(C–D)}{2}$, we get: $= 2\sin\frac{(6x+2x)}{2}\cos\frac{(6x–2x)}{2}+2\sin4x$ $= 2\sin4x\cos2x+2\sin4x$ $=2\sin4x(\cos2x+1)$ Using formula: $2\cos^2x=\cos2x+1$, we get: $=2\sin4x(2\cos^2x)$ $= 4\cos^2x\sin4x$ Hence, the correct answer is $4\cos^2x\sin4x$.
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