The value of `2 sin 3x cos 2x` is equal to

Options

A. `sin 5x+ sin x`
B. `sin 3x+ sin x`
C. `sin 7x + sin x`
D. `sin 4x + sin x`

Answer

To find the value of \(2 \sin 3x \cos 2x\), we can use the trigonometric identity for the product of sine and cosine. The identity states:

\[ 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \]

In our case, we have:

- \(A = 3x\)
- \(B = 2x\)

Now, we can apply the identity:

\[ 2 \sin 3x \cos 2x = \sin(3x + 2x) + \sin(3x - 2x) \]

Now, let's simplify the expressions inside the sine functions:

1. Calculate \(3x + 2x\):
\[ 3x + 2x = 5x \]

2. Calculate \(3x - 2x\):
\[ 3x - 2x = x \]

Putting it all together, we have:

\[ 2 \sin 3x \cos 2x = \sin(5x) + \sin(x) \]

Thus, the value of \(2 \sin 3x \cos 2x\) is:

\[ \sin(5x) + \sin(x) \]

### Final Answer:
\[ 2 \sin 3x \cos 2x = \sin(5x) + \sin(x) \]