Tan[2*tan⁻¹⅕-∏/4]

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To find the value of tan[2*tan⁻¹(1/5) - ∏/4], we can start by using the inverse tangent function to find the angle whose tangent is 1/5.

tan⁻¹(1/5) represents the angle whose tangent is 1/5. Let's call this angle θ.

Now, using the tangent addition formula, which states that tan(A + B) = (tan A + tan B) / (1 - tan A * tan B), we can find the value of tan[2*tan⁻¹(1/5) - ∏/4].

Step 1: Find tan(θ)
Given that tan(θ) = 1/5, we can solve for θ using the inverse tangent function.
θ = tan⁻¹(1/5) ≈ 11.31°

Step 2: Use the tangent addition formula
Let A = 2*tan⁻¹(1/5) and B = -∏/4.
tan(A) = tan[2*tan⁻¹(1/5)]
tan(B) = tan(-∏/4)

Using the tangent addition formula:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)

We substitute the values into the formula:
tan[2*tan⁻¹(1/5) - ∏/4] = (tan(2 * tan⁻¹(1/5)) + tan(-∏/4)) / (1 - tan(2 * tan⁻¹(1/5)) * tan(-∏/4))

Step 3: Evaluate the formula
tan(2 * tan⁻¹(1/5)) = tan(2 * 11.31°) ≈ tan(22.62°) ≈ 0.4142
tan(-∏/4) = -1

Now we substitute the values back into the formula:
tan[2*tan⁻¹(1/5) - ∏/4] = (0.4142 + (-1)) / (1 - 0.4142 * (-1))
= (-0.5858) / (1 + 0.4142)
= -0.5858 / 1.4142
≈ -0.4142

Therefore, tan[2*tan⁻¹(1/5) - ∏/4] is approximately equal to -0.4142.