The expression `2"cos"(pi)/(17)*"cos"(9pi)/(17)+"cos"(7pi)/(17)+"cos"(9pi)/(17)` simplifies to an integer P. Find the value of P.

Answer

To solve the expression \( P = 2 \cos\left(\frac{\pi}{17}\right) \cos\left(\frac{9\pi}{17}\right) + \cos\left(\frac{7\pi}{17}\right) + \cos\left(\frac{9\pi}{17}\right) \), we can simplify it step by step.

### Step 1: Rewrite the Expression
We start with the expression:
\[ P = 2 \cos\left(\frac{\pi}{17}\right) \cos\left(\frac{9\pi}{17}\right) + \cos\left(\frac{7\pi}{17}\right) + \cos\left(\frac{9\pi}{17}\right) \]

### Step 2: Apply the Product-to-Sum Formula
We can use the formula \( 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \). Here, let \( A = \frac{\pi}{17} \) and \( B = \frac{9\pi}{17} \):
\[ 2 \cos\left(\frac{\pi}{17}\right) \cos\left(\frac{9\pi}{17}\right) = \cos\left(\frac{\pi}{17} + \frac{9\pi}{17}\right) + \cos\left(\frac{\pi}{17} - \frac{9\pi}{17}\right) \]
Calculating the angles:
\[ \frac{\pi}{17} + \frac{9\pi}{17} = \frac{10\pi}{17} \]
\[ \frac{\pi}{17} - \frac{9\pi}{17} = -\frac{8\pi}{17} \]
Using the property \( \cos(-x) = \cos(x) \):
\[ \cos\left(-\frac{8\pi}{17}\right) = \cos\left(\frac{8\pi}{17}\right) \]
Thus, we have:
\[ 2 \cos\left(\frac{\pi}{17}\right) \cos\left(\frac{9\pi}{17}\right) = \cos\left(\frac{10\pi}{17}\right) + \cos\left(\frac{8\pi}{17}\right) \]

### Step 3: Substitute Back into the Expression
Now substituting back into \( P \):
\[ P = \cos\left(\frac{10\pi}{17}\right) + \cos\left(\frac{8\pi}{17}\right) + \cos\left(\frac{7\pi}{17}\right) + \cos\left(\frac{9\pi}{17}\right) \]

### Step 4: Group the Cosine Terms
Notice that we can group the terms:
\[ P = \left(\cos\left(\frac{10\pi}{17}\right) + \cos\left(\frac{7\pi}{17}\right)\right) + \left(\cos\left(\frac{8\pi}{17}\right) + \cos\left(\frac{9\pi}{17}\right)\right) \]

### Step 5: Apply the Sum-to-Product Formula
Using the sum-to-product formula \( \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \):
1. For \( \cos\left(\frac{10\pi}{17}\right) + \cos\left(\frac{7\pi}{17}\right) \):
- \( A = \frac{10\pi}{17}, B = \frac{7\pi}{17} \)
- \( \frac{A + B}{2} = \frac{17\pi}{34} = \frac{\pi}{2} \)
- \( \frac{A - B}{2} = \frac{3\pi}{34} \)
- This gives \( 2 \cos\left(\frac{\pi}{2}\right) \cos\left(\frac{3\pi}{34}\right) = 0 \) (since \( \cos\left(\frac{\pi}{2}\right) = 0 \)).

2. For \( \cos\left(\frac{8\pi}{17}\right) + \cos\left(\frac{9\pi}{17}\right) \):
- \( A = \frac{8\pi}{17}, B = \frac{9\pi}{17} \)
- \( \frac{A + B}{2} = \frac{17\pi}{34} = \frac{\pi}{2} \)
- \( \frac{A - B}{2} = -\frac{\pi}{34} \)
- This also gives \( 2 \cos\left(\frac{\pi}{2}\right) \cos\left(-\frac{\pi}{34}\right) = 0 \).

### Final Step: Conclusion
Since both pairs sum to zero:
\[ P = 0 + 0 = 0 \]

Thus, the value of \( P \) is:
\[ \boxed{0} \]