Let `P(k)=(1+cos(pi/(4k)))` `(1+cos(((2k-1)pi)/(4k)))` `(1+cos(((2k+1)pi)/(4k)))(1+cos(((4k-1)pi)/(4k)))dot` Then Prove that (a)`P(3)=1/(16)` (b) `P(4)=(2-sqrt(2))/(16)` (c) `P(5)=(3-sqrt(5))/(32)` (d) `P(6)(2-sqrt(3))/(16)`

Explanation

To solve the problem, we need to evaluate the function \( P(k) = (1 + \cos(\frac{\pi}{4k})) (1 + \cos(\frac{(2k-1)\pi}{4k})) (1 + \cos(\frac{(2k+1)\pi}{4k})) (1 + \cos(\frac{(4k-1)\pi}{4k})) \) for \( k = 3, 4, 5, \) and \( 6 \).

### Step-by-Step Solution:

1. **Evaluate \( P(3) \)**:
\[ P(3) = (1 + \cos(\frac{\pi}{12})) (1 + \cos(\frac{5\pi}{12})) (1 + \cos(\frac{7\pi}{12})) (1 + \cos(\frac{11\pi}{12})) \]
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