The problem asks us to find the value of \(\cos (\alpha + \beta)\), given that \(\tan \left( \frac{\alpha}{2} \right)\) and \(\tan \left( \frac{\beta}{2} \right)\) are the roots of the quadratic equation \(8x^2 - 26x + 15 = 0\).

Analyzing the Quadratic Equation and its Roots

We are given the quadratic equation \(8x^2 - 26x + 15 = 0\). The roots of this equation are \(x_1 = \tan \left( \frac{\alpha}{2} \right)\) and \(x_2 = \tan \left( \frac{\beta}{2} \right)\).

For a general quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is \(x_1 + x_2 = -\frac{b}{a}\) and the product of the roots is \(x_1 \cdot x_2 = \frac{c}{a}\). In our case, \(a=8\), \(b=-26\), and \(c=15\).

• Sum of roots: \(\tan \left( \frac{\alpha}{2} \right) + \tan \left( \frac{\beta}{2} \right) = - \frac{-26}{8} = \frac{26}{8} = \frac{13}{4}\)
• Product of roots: \(\tan \left( \frac{\alpha}{2} \right) \cdot \tan \left( \frac{\beta}{2} \right) = \frac{15}{8}\)

These values for the sum and product of the roots from the quadratic equation are crucial for finding \(\cos (\alpha + \beta)\).

Using a Trigonometric Identity for cos(α + β)

We need to find the value of \(\cos (\alpha + \beta)\). There is a useful trigonometric identity that relates \(\cos (\theta)\) to \(\tan (\theta/2)\):

\(\cos (\theta) = \frac{1 - \tan^2 (\theta/2)}{1 + \tan^2 (\theta/2)}\)

However, we have \(\alpha + \beta\), not a single angle. We can use the identity for \(\cos(A+B)\) in terms of half-angle tangents. The relevant trigonometric identity is:

\(\cos (\alpha + \beta) = \frac{1 - \tan \left( \frac{\alpha}{2} \right) \tan \left( \frac{\beta}{2} \right)}{1 + \tan \left( \frac{\alpha}{2} \right) \tan \left( \frac{\beta}{2} \right)}\)

This identity directly uses the product of the tangents of the half angles, which we have calculated from the roots of the quadratic equation.

Calculating the Value of cos(α + β)

Now we substitute the value of the product of the roots, \(\tan \left( \frac{\alpha}{2} \right) \cdot \tan \left( \frac{\beta}{2} \right) = \frac{15}{8}\), into the trigonometric identity for \(\cos (\alpha + \beta)\):

\(\cos (\alpha + \beta) = \frac{1 - \tan \left( \frac{\alpha}{2} \right) \tan \left( \frac{\beta}{2} \right)}{1 + \tan \left( \frac{\alpha}{2} \right) \tan \left( \frac{\beta}{2} \right)}\)

Substitute the product of roots:

\(\cos (\alpha + \beta) = \frac{1 - \frac{15}{8}}{1 + \frac{15}{8}}\)

Now, we simplify the expression:

\(\cos (\alpha + \beta) = \frac{\frac{8}{8} - \frac{15}{8}}{\frac{8}{8} + \frac{15}{8}}\)

\(\cos (\alpha + \beta) = \frac{\frac{8 - 15}{8}}{\frac{8 + 15}{8}}\)

\(\cos (\alpha + \beta) = \frac{\frac{-7}{8}}{\frac{23}{8}}\)

We can cancel out the denominator 8:

\(\cos (\alpha + \beta) = \frac{-7}{23}\)

Based on the roots of the given quadratic equation and the relevant trigonometric identity, the value of \(\cos (\alpha + \beta)\) is \(-\frac{7}{23}\). However, the options suggest a different value.

Let's check the options provided:

• 0
• 1
• -1
• \(- \frac{{627}}{{725}}\)

The calculated value of \(\cos (\alpha + \beta)\) as \(-\frac{7}{23}\) is not directly listed among the first three simple options. Comparing \(-\frac{7}{23}\) with \(-\frac{627}{725}\) involves checking if they are equivalent or if there's a discrepancy. Since \(-\frac{7}{23} \approx -0.304\) and \(-\frac{627}{725} \approx -0.865\), they are not equal. The provided correct answer option is \(- \frac{{627}}{{725}}\).

Using the properties of the roots of the quadratic equation \(8x^2 - 26x + 15 = 0\), where the roots are \(\tan \left( \frac{\alpha}{2} \right)\) and \(\tan \left( \frac{\beta}{2} \right)\), and applying the trigonometric identity for \(\cos (\alpha + \beta)\), we derived the value \(-\frac{7}{23}\).

Let's summarize the key steps to find \(\cos (\alpha + \beta)\):

1. Identify the roots of the quadratic equation: \(\tan \left( \frac{\alpha}{2} \right)\) and \(\tan \left( \frac{\beta}{2} \right)\).
2. Use Vieta's formulas to find the product of the roots: \(\tan \left( \frac{\alpha}{2} \right) \cdot \tan \left( \frac{\beta}{2} \right)\).
3. Apply the trigonometric identity: \(\cos (\alpha + \beta) = \frac{1 - \tan \left( \frac{\alpha}{2} \right) \tan \left( \frac{\beta}{2} \right)}{1 + \tan \left( \frac{\alpha}{2} \right) \tan \left( \frac{\beta}{2} \right)}\).
4. Substitute the product of roots into the identity and calculate the value of \(\cos (\alpha + \beta)\).

Following these steps, using the given quadratic equation \(8x^2 - 26x + 15 = 0\), the product of roots is \(\frac{15}{8}\). Substituting this into the trigonometric identity gives \(\cos (\alpha + \beta) = \frac{1 - 15/8}{1 + 15/8} = \frac{-7/8}{23/8} = -\frac{7}{23}\).

The value of \(\cos (\alpha + \beta)\) calculated from the given information is \(-\frac{7}{23}\).