What is the value of [sin (y – z) + sin (y + z) + 2 sin y]/[sin (x – z) + sin (x + z) + 2 sin x]?

Options

A. cosx siny
B. (siny) / (sinx)
C. sinz
D. sinx tany

Answer

To solve the problem, we need to evaluate the expression:

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

### Step 1: Simplify the Numerator
Using the sine addition formula, we can simplify the numerator:

\[ \sin(y - z) + \sin(y + z) = 2 \sin y \cos z \]

So, the numerator becomes:

\[ 2 \sin y \cos z + 2 \sin y = 2 \sin y (\cos z + 1) \]

### Step 2: Simplify the Denominator
Now, we apply the same sine addition formula to the denominator:

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

Thus, the denominator becomes:

\[ 2 \sin x \cos z + 2 \sin x = 2 \sin x (\cos z + 1) \]

### Step 3: Substitute Back into the Expression
Now we can substitute the simplified numerator and denominator back into the expression:

\[ \frac{2 \sin y (\cos z + 1)}{2 \sin x (\cos z + 1)} \]

### Step 4: Cancel Common Terms
We can cancel the common terms \(2\) and \((\cos z + 1)\) from the numerator and denominator (assuming \(\cos z + 1 \neq 0\)):

\[ \frac{\sin y}{\sin x} \]

### Final Result
Thus, the value of the given expression is:

\[ \frac{\sin y}{\sin x} \]