The value of the expression
`(sin7alpha+6sin 5 alpha+17sin3alpha+12sin alpha)/(sin6alpha+5sin 4alpha+12 sin 2 alpha)`, where `alpha=(pi)/(5)` is equal to :

Options

A. `(sqrt(5)-1)/(4)`
B. ` (sqrt(5)+1)/(4)`
C. `(sqrt(5)+1)/(2)`
D. `(sqrt(5)-1)/(2)`

Answer

To solve the expression

\[ \frac{\sin(7\alpha) + 6\sin(5\alpha) + 17\sin(3\alpha) + 12\sin(\alpha)}{\sin(6\alpha) + 5\sin(4\alpha) + 12\sin(2\alpha)} \]

where \(\alpha = \frac{\pi}{5}\), we will follow these steps:

### Step 1: Substitute \(\alpha\)
First, substitute \(\alpha\) with \(\frac{\pi}{5}\):

\[ \sin(7\alpha) = \sin\left(7 \cdot \frac{\pi}{5}\right) = \sin\left(\frac{7\pi}{5}\right) \]
\[ \sin(5\alpha) = \sin\left(5 \cdot \frac{\pi}{5}\right) = \sin(\pi) = 0 \]
\[ \sin(3\alpha) = \sin\left(3 \cdot \frac{\pi}{5}\right) \]
\[ \sin(\alpha) = \sin\left(\frac{\pi}{5}\right) \]

### Step 2: Simplify the Numerator
The numerator simplifies to:

\[ \sin\left(\frac{7\pi}{5}\right) + 6(0) + 17\sin\left(\frac{3\pi}{5}\right) + 12\sin\left(\frac{\pi}{5}\right) \]

Using the property \(\sin\left(\frac{7\pi}{5}\right) = -\sin\left(\frac{2\pi}{5}\right)\):

\[ -\sin\left(\frac{2\pi}{5}\right) + 17\sin\left(\frac{3\pi}{5}\right) + 12\sin\left(\frac{\pi}{5}\right) \]

### Step 3: Simplify the Denominator
Now, simplify the denominator:

\[ \sin(6\alpha) = \sin\left(6 \cdot \frac{\pi}{5}\right) = \sin\left(\frac{6\pi}{5}\right) \]
\[ \sin(4\alpha) = \sin\left(4 \cdot \frac{\pi}{5}\right) \]
\[ \sin(2\alpha) = \sin\left(2 \cdot \frac{\pi}{5}\right) \]

The denominator becomes:

\[ \sin\left(\frac{6\pi}{5}\right) + 5\sin\left(\frac{4\pi}{5}\right) + 12\sin\left(\frac{2\pi}{5}\right) \]

Using the property \(\sin\left(\frac{6\pi}{5}\right) = -\sin\left(\frac{\pi}{5}\right)\):

\[ -\sin\left(\frac{\pi}{5}\right) + 5\sin\left(\frac{4\pi}{5}\right) + 12\sin\left(\frac{2\pi}{5}\right) \]

### Step 4: Substitute Values
Now we can substitute the known values:

1. \(\sin\left(\frac{\pi}{5}\right)\)
2. \(\sin\left(\frac{2\pi}{5}\right)\)
3. \(\sin\left(\frac{3\pi}{5}\right) = \sin\left(\pi - \frac{2\pi}{5}\right) = \sin\left(\frac{2\pi}{5}\right)\)
4. \(\sin\left(\frac{4\pi}{5}\right) = \sin\left(\pi - \frac{\pi}{5}\right) = \sin\left(\frac{\pi}{5}\right)\)

### Step 5: Final Expression
After substituting these values, we can simplify the expression further.

The numerator becomes:

\[ -\sin\left(\frac{2\pi}{5}\right) + 17\sin\left(\frac{2\pi}{5}\right) + 12\sin\left(\frac{\pi}{5}\right) = 16\sin\left(\frac{2\pi}{5}\right) + 12\sin\left(\frac{\pi}{5}\right) \]

The denominator simplifies to:

\[ -\sin\left(\frac{\pi}{5}\right) + 5\sin\left(\frac{\pi}{5}\right) + 12\sin\left(\frac{2\pi}{5}\right) = 4\sin\left(\frac{\pi}{5}\right) + 12\sin\left(\frac{2\pi}{5}\right) \]

### Step 6: Cancel Common Terms
Now we can cancel out the common terms in the numerator and denominator:

\[ \frac{16\sin\left(\frac{2\pi}{5}\right) + 12\sin\left(\frac{\pi}{5}\right)}{4\sin\left(\frac{\pi}{5}\right) + 12\sin\left(\frac{2\pi}{5}\right)} \]

### Step 7: Final Calculation
This will reduce to a simpler form, and substituting the known values of \(\sin\left(\frac{\pi}{5}\right)\) and \(\sin\left(\frac{2\pi}{5}\right)\) will yield the final answer.

### Final Answer
After performing the calculations, we find that the value of the expression is:

\[ \frac{\sqrt{5}+1}{2} \]