Introduction:
In this problem, we are required to find the value of tan(2tan^-1(1/5) - π/4). We can solve this problem using trigonometric identities and properties.

Solution:
To solve this problem, we will follow the below steps.

Step 1: Find the value of tan^-1(1/5)
We know that tan(tan^-1(x)) = x. Using this property, we can find the value of tan^-1(1/5) as follows:

tan(tan^-1(1/5)) = 1/5
tan^-1(1/5) = tan^-1(tan(atan(1/5))) = atan(1/5)

So, the value of tan^-1(1/5) is atan(1/5).

Step 2: Find the value of 2tan^-1(1/5) - π/4
We can use the identity tan(2θ) = (2tanθ)/(1-tan^2θ) to find the value of tan(2tan^-1(1/5)).

tan(2tan^-1(1/5)) = (2tan(tan^-1(1/5)))/(1-tan^2(tan^-1(1/5)))
= (2(1/5))/(1-(1/5)^2)
= 10/24

Now, we can find the value of 2tan^-1(1/5) - π/4 as follows:

2tan^-1(1/5) - π/4 = 2atan(1/5) - π/4

Step 3: Find the value of tan(2tan^-1(1/5) - π/4)
We can use the identity tan(a-b) = (tan(a) - tan(b))/(1+tan(a)tan(b)) to find the value of tan(2tan^-1(1/5) - π/4).

tan(2tan^-1(1/5) - π/4) = (tan(2tan^-1(1/5)) - tan(π/4))/(1+tan(2tan^-1(1/5))tan(π/4))
= ((10/24) - 1)/(1+(10/24))
= -7/17

Conclusion:
Therefore, the value of tan(2tan^-1(1/5) - π/4) is -7/17.