Evaluate each of the following:

• (i) tan( tan⁻¹(1/3) / 4 )
• (ii) tan( sin⁻¹(1/5) / 13 )
• (iii) tan( cos⁻¹(1/8) / 17 )

To evaluate the expressions involving the tangent and inverse trigonometric functions, we can break them down step by step.

1. Evaluating tan(tan⁻¹(1/3) / 4)

Let’s start with the first expression:

Let θ = tan⁻¹(1/3). This means that tan(θ) = 1/3. We want to find:

tan(θ / 4). Using the tangent half-angle formula:

• tan(θ / 4) = sin(θ / 4) / cos(θ / 4)

Using the known value of tan(θ), we can find sin(θ) and cos(θ) using the identity:

• sin(θ) = 1/√(1 + (1/3)²) = 1/√(10/9) = 3/√10
• cos(θ) = 3/√(10/9) = 9/√10

Now, we can apply the tangent formula for small angles to find tan(θ / 4).

2. Evaluating tan(sin⁻¹(1/5) / 13)

Next, we consider:

Let φ = sin⁻¹(1/5). This implies sin(φ) = 1/5. We need to find:

tan(φ / 13). Using the identity:

• cos(φ) = √(1 - sin²(φ)) = √(1 - (1/5)²) = √(24/25) = 2√6/5

Then, we can find tan(φ) = sin(φ) / cos(φ) = (1/5) / (2√6/5) = 1/(2√6).

Now, we can use the tangent formula for small angles to find tan(φ / 13).

3. Evaluating tan(cos⁻¹(1/8) / 17)

Finally, we evaluate:

Let ψ = cos⁻¹(1/8). This means cos(ψ) = 1/8. We want to find:

tan(ψ / 17). Using the identity:

• sin(ψ) = √(1 - cos²(ψ)) = √(1 - (1/8)²) = √(63/64) = √63/8

Thus, tan(ψ) = sin(ψ) / cos(ψ) = (√63/8) / (1/8) = √63.

Now, we can apply the tangent formula for small angles to find tan(ψ / 17).

In summary, each expression can be evaluated using trigonometric identities and properties of inverse functions. The final values will depend on the calculations performed for each step.