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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