If `sin(x+3 alpha)=3sin (alpha-x)` then

Options

A. `tanx=tanalpha`
B. `tanx=tan^(2)alpha`
C. `tanx=tan^(3)alpha`
D. `tanx=3tan alpha`

Explanation

To solve the equation \( \sin(x + 3\alpha) = 3\sin(\alpha - x) \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sin(x + 3\alpha) = 3\sin(\alpha - x) \]

### Step 2: Use the sine addition formula
Using the sine addition formula, we can expand \( \sin(x + 3\alpha) \):
\[ \sin(x + 3\alpha) = \sin x \cos(3\alpha) + \cos x \sin(3\alpha) \]

For \( \sin(3\alpha) \), we can use the triple angle formula:
\[ \sin(3\alpha) = 3\sin(\alpha) - 4\sin^3(\alpha) \]
And for \( \cos(3\alpha) \), we can use:
\[ \cos(3\alpha) = 4\cos^3(\alpha) - 3\cos(\alpha) \]

### Step 3: Expand the right side
Now we expand the right side \( 3\sin(\alpha - x) \) using the sine subtraction formula:
\[ \sin(\alpha - x) = \sin(\alpha)\cos(x) - \cos(\alpha)\sin(x) \]
Thus,
\[ 3\sin(\alpha - x) = 3\sin(\alpha)\cos(x) - 3\cos(\alpha)\sin(x) \]

### Step 4: Set up the equation
Now we can set up the equation:
\[ \sin x (4\cos^3(\alpha) - 3\cos(\alpha)) + \cos x (3\sin(\alpha) - 4\sin^3(\alpha)) = 3\sin(\alpha)\cos(x) - 3\cos(\alpha)\sin(x) \]

### Step 5: Rearranging terms
Rearranging the terms gives us:
\[ \sin x (4\cos^3(\alpha) - 3\cos(\alpha) + 3\cos(\alpha)) = 3\sin(\alpha)\cos(x) - \cos x(3\sin(\alpha) - 4\sin^3(\alpha)) \]
This simplifies to:
\[ \sin x (4\cos^3(\alpha)) = 3\sin(\alpha)\cos(x) - \cos x(3\sin(\alpha) - 4\sin^3(\alpha)) \]

### Step 6: Factor out common terms
We can factor out common terms and simplify further. This will lead us to a relationship between \( \tan x \) and \( \tan \alpha \).

### Step 7: Solve for \( \tan x \)
After simplification, we find that:
\[ \tan x = \tan^3 \alpha \]

### Conclusion
Thus, we conclude that:
\[ \tan x = \tan^3 \alpha \]