Solution:

Given equation: x^4 - 8x^3 + ax^2 - bx + 16 = 0

To find the values of x1, x2, x3, x4, we can use the concept of Vieta's formulas. Vieta's formulas state that for a polynomial equation of the form:

ax^n + bx^(n-1) + cx^(n-2) + ... + k = 0

The sum of the roots is equal to the negation of the coefficient of the second highest power term (b/a) and the product of the roots is equal to the constant term divided by the coefficient of the highest power term (k/a).

In this case, the coefficient of the second highest power term is -8 and the constant term is 16. Therefore, we have:

Sum of the roots, x1 + x2 + x3 + x4 = -(-8) = 8 ...(1)
Product of the roots, x1 * x2 * x3 * x4 = 16 ...(2)

Now, we need to find the value of tan^(-1)(x1), tan^(-1)(x2), tan^(-1)(x3), and tan^(-1)(x4).

Using the identity:
tan^(-1)(a) + tan^(-1)(b) = tan^(-1)[(a + b) / (1 - ab)]

We can find the value of tan^(-1)(x1 + x2), tan^(-1)(x3 + x4), and tan^(-1)(x1 + x2 + x3 + x4) using the sum of the roots and then use the above identity to find the values of tan^(-1)(x1), tan^(-1)(x2), tan^(-1)(x3), and tan^(-1)(x4).

Now let's solve it step by step:

Step 1: Find the values of x1 + x2, x3 + x4, and x1 + x2 + x3 + x4.

Using Vieta's formulas, we know that x1 + x2 + x3 + x4 = 8 ...(1)

Step 2: Find the values of tan^(-1)(x1 + x2), tan^(-1)(x3 + x4), and tan^(-1)(x1 + x2 + x3 + x4).

Using the identity: tan^(-1)(a) + tan^(-1)(b) = tan^(-1)[(a + b) / (1 - ab)]

tan^(-1)(x1 + x2) = tan^(-1)[(x1 + x2) / (1 - (x1)(x2))] ...(3)
tan^(-1)(x3 + x4) = tan^(-1)[(x3 + x4) / (1 - (x3)(x4))] ...(4)
tan^(-1)(x1 + x2 + x3 + x4) = tan^(-1)[(x1 + x2 + x3 + x4) / (1 - (x1 + x2)(x3 + x4))] ...(5)