What is the value of `{(sin4x+sin4y)tan(2x-2y)}/(sin4x-sin4y)?`

Options

A. `tan 2 (2x + 2y) `
B. `tan(2)`
C. `cot(x - y)`
D. `tan (2x + 2y)`

Answer

To solve the expression \(\frac{\sin 4x + \sin 4y}{\sin 4x - \sin 4y} \tan(2x - 2y)\), we can utilize the sum-to-product identities for sine functions. Here’s a step-by-step breakdown of the solution:

### Step 1: Apply the sum-to-product identities
We know the following identities:
- \(\sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right)\)
- \(\sin a - \sin b = 2 \cos\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right)\)

Using these identities, we can rewrite \(\sin 4x + \sin 4y\) and \(\sin 4x - \sin 4y\):

\[ \sin 4x + \sin 4y = 2 \sin\left(2x + 2y\right) \cos\left(2x - 2y\right) \]
\[ \sin 4x - \sin 4y = 2 \cos\left(2x + 2y\right) \sin\left(2x - 2y\right) \]

### Step 2: Substitute into the expression
Now substitute these identities into the original expression:

\[ \frac{\sin 4x + \sin 4y}{\sin 4x - \sin 4y} = \frac{2 \sin(2x + 2y) \cos(2x - 2y)}{2 \cos(2x + 2y) \sin(2x - 2y)} \]

### Step 3: Simplify the expression
The \(2\) in the numerator and denominator cancels out:

\[ = \frac{\sin(2x + 2y) \cos(2x - 2y)}{\cos(2x + 2y) \sin(2x - 2y)} \]

### Step 4: Use the definition of tangent
We can express this as:

\[ = \tan(2x + 2y) \cdot \frac{\cos(2x - 2y)}{\sin(2x - 2y)} \]

The term \(\frac{\cos(2x - 2y)}{\sin(2x - 2y)}\) is equal to \(\cot(2x - 2y)\):

\[ = \tan(2x + 2y) \cdot \cot(2x - 2y) \]

### Step 5: Final simplification
Since \(\cot(2x - 2y) = \frac{1}{\tan(2x - 2y)}\), we can rewrite it as:

\[ = \tan(2x + 2y) \cdot \frac{1}{\tan(2x - 2y)} \cdot \tan(2x - 2y) \]

This simplifies to:

\[ = \tan(2x + 2y) \]

### Conclusion
Thus, the value of the expression \(\frac{\sin 4x + \sin 4y}{\sin 4x - \sin 4y} \tan(2x - 2y)\) is:

\[ \boxed{\tan(2x + 2y)} \]