8 sin(x/8) cos(x/2) cos(x/4) cos(x/8) is Equal to What?
Question
\(8\sin\frac{x}{8}\cos\frac{x}{2}\cos\frac{x}{4}\cos\frac{x}{8}\) is equal to
(a) \(8\cos x\)
(b) \(\cos x\)
(c) \(8\sin x\)
(d) \(\sin x\)
Solution
Use the identity:
\[ 2\sin A\cos A=\sin 2A \]
\[ 8\sin\frac{x}{8}\cos\frac{x}{8} =4\sin\frac{x}{4} \]
\[ 4\sin\frac{x}{4}\cos\frac{x}{4} =2\sin\frac{x}{2} \]
\[ 2\sin\frac{x}{2}\cos\frac{x}{2} =\sin x \]
Therefore,
\[ 8\sin\frac{x}{8}\cos\frac{x}{2}\cos\frac{x}{4}\cos\frac{x}{8} =\sin x \]
Final Answer
\[ \boxed{\sin x} \]
Hence, the correct option is (d) \(\sin x\).