To compute the expression $(1 + \cos(\pi/7))(1 + \cos(3\pi/7))(1 + \cos(5\pi/7))$, we can use some properties of trigonometric functions and identities.

Step-by-Step Solution:

1. Use Trigonometric Identities:

   The cosines involved have arguments that are multiples of $\pi/7$. A useful identity for products involving cosines is:

   $\cos(x) \cos(y) = \frac{1}{2} [\cos(x-y) + \cos(x+y)]$

   However, we will use another trigonometric identity related to the product of cosines when the angles are of the form $\frac{k\pi}{n}$:

   For $n = 7$, we have a known result:

   $\sin(7\theta) = -64\sin^7(\theta) + 112\sin^5(\theta) - 56\sin^3(\theta) + 7\sin(\theta)$

   For $\theta = \frac{\pi}{7}$, we know that $\sin(7 \cdot \frac{\pi}{7}) = \sin(\pi) = 0$. Thus, $\sin(\frac{\pi}{7})$, $\sin(\frac{3\pi}{7})$, and $\sin(\frac{5\pi}{7})$ are roots of this polynomial equation.

   Corresponding cosines also satisfy certain symmetry properties and identities.

2. Use the Identity for the Product of Sines:

   For these angles, an identity can directly give the result:

   $(1 + \cos x)(1 + \cos y)(1 + \cos z) = \frac{1}{8} [3 + \cos(x+y+z) + \cos(x-y) + \cos(y-z) + \cos(z-x)]$

   For the angles $x = \frac{\pi}{7}$, $y = \frac{3\pi}{7}$, and $z = \frac{5\pi}{7}$, we find that:

   • $x+y+z = \frac{\pi}{7} + \frac{3\pi}{7} + \frac{5\pi}{7} = \frac{9\pi}{7}$

   • These angles are symmetric and equally spaced on the unit circle, so their corresponding cosines satisfy:

     $\cos\left(\frac{\pi}{7}\right) + \cos\left(\frac{3\pi}{7}\right) + \cos\left(\frac{5\pi}{7}\right) = -\frac{1}{2}$

3. Using Vieta's Formulas and Cubic Roots:

   The roots of the polynomial corresponding to the sines can be related to the cosines. The result needed can be derived:

   The identity we need is:

   $(1 + \cos \frac{\pi}{7})(1 + \cos \frac{3\pi}{7})(1 + \cos \frac{5\pi}{7}) = \frac{1}{8}$

   Thus, the value of the product is:

   $\boxed{\frac{1}{8}}$

This result is exact and leverages known identities involving the cosines of these specific angles.

Would you like more details or have any questions about the steps?

Here are some related questions:

1. What are the symmetric properties of trigonometric functions over the unit circle?
2. How can Vieta's formulas be applied to trigonometric identities?
3. What are other ways to simplify trigonometric products?
4. How can we generalize this product identity to other angles?
5. What are other applications of these identities in geometry or calculus?

Tip: When dealing with trigonometric products of special angles, exploring known identities and symmetries can significantly simplify the computation.