1) Write $$ 4 \sin 2 x \cos 2 x \cos 4 x $$ as a single sine. a) $$ \sin 16 x $$ b) $$ \sin 2 x $$ c) $$ \sin 4 x $$ d) $$ \sin 8 x $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$ 4 \sin 2x \cos 2x \cos 4x $$, we can use the double angle identity for sine, which states that:

$$
\sin 2\theta = 2 \sin \theta \cos \theta
$$

First, we can apply this identity to $$ 4 \sin 2x \cos 2x $$:

$$
4 \sin 2x \cos 2x = 2 \cdot 2 \sin 2x \cos 2x = 2 \sin 4x
$$

Now, we can rewrite the original expression:

$$
4 \sin 2x \cos 2x \cos 4x = 2 \sin 4x \cos 4x
$$

Next, we apply the double angle identity again to $$ 2 \sin 4x \cos 4x $$:

$$
2 \sin 4x \cos 4x = \sin 8x
$$

Thus, we have:

$$
4 \sin 2x \cos 2x \cos 4x = \sin 8x
$$

Therefore, the answer is:

$$
\boxed{\sin 8x}
$$
$$ 4 \sin 2x \cos 2x \cos 4x = \sin 8x $$

1) Write \( 4 \sin 2 x \cos 2 x \cos 4 x \) as a single sine. a) \( \sin 16 x \) b) \( \sin 2 x \) c) \( \sin 4 x \) d) \( \sin 8 x \)