$$ \sin (x)\cos (x)\cos (2x)\cos (4x)=\frac{1}{8}\sin (8x) $$

Question

UpStudy AI · Accepted Answer

1. Start with the given expression:
   
   $$
   \sin (x)\cos (x)\cos (2x)\cos (4x)
   $$

2. Recognize that:

$$
   \sin (x)\cos (x)=\frac{1}{2}\sin (2x)
   $$
   
   Therefore, the expression becomes:

$$
   \frac{1}{2}\sin (2x)\cos (2x)\cos (4x)
   $$

3. Notice that:

$$
   \sin (2x)\cos (2x)=\frac{1}{2}\sin (4x)
   $$
   
   Substituting, we have:

$$
   \frac{1}{2}\cdot\frac{1}{2}\sin (4x)\cos (4x)=\frac{1}{4}\sin (4x)\cos (4x)
   $$

4. Finally, use the identity:

$$
   \sin (4x)\cos (4x)=\frac{1}{2}\sin (8x)
   $$
   
   Which gives:

$$
   \frac{1}{4}\cdot\frac{1}{2}\sin (8x)=\frac{1}{8}\sin (8x)
   $$

Thus, the identity is proven:

$$
\sin (x)\cos (x)\cos (2x)\cos (4x)=\frac{1}{8}\sin (8x)
$$
$$ \sin (x)\cos (x)\cos (2x)\cos (4x) = \frac{1}{8}\sin (8x) $$

\( \sin (x)\cos (x)\cos (2x)\cos (4x)=\frac{1}{8}\sin (8x) \)