\(2(1-2\sin^2 7x)\sin 3x\) is Equal to What?
Question
Find the value of
\[ 2(1-2\sin^2 7x)\sin 3x \]
(a) \(\sin17x-\sin11x\)
(b) \(\sin11x-\sin17x\)
(c) \(\cos17x-\cos11x\)
(d) \(\cos17x+\cos11x\)
Solution
Using the identity
\[ 1-2\sin^2\theta=\cos2\theta \]
we get
\[ 2(1-2\sin^2 7x)\sin3x = 2\cos14x\sin3x \]
Now use
\[ 2\sin A\cos B = \sin(A+B)+\sin(A-B) \]
with \[ A=3x,\quad B=14x \]
\[ 2\cos14x\sin3x = \sin(17x)+\sin(-11x) \]
Since
\[ \sin(-11x)=-\sin11x \]
therefore
\[ 2(1-2\sin^2 7x)\sin3x = \sin17x-\sin11x \]
Final Answer
\[ \boxed{\sin17x-\sin11x} \]
Hence, the correct option is (a) \(\sin17x-\sin11x\).