**Proof:**

To prove that (sec 8θ - 1)/(sec 4θ - 1) = (tan 8θ)/(tan 4θ), we will manipulate the left-hand side of the equation to match the right-hand side.

**Step 1: Express everything in terms of sine and cosine**

We know that sec θ = 1/cos θ and tan θ = sin θ / cos θ. So, let's express the given equation in terms of sine and cosine:

(sec 8θ - 1)/(sec 4θ - 1) = (tan 8θ)/(tan 4θ)

(1/cos 8θ - 1)/(1/cos 4θ - 1) = (sin 8θ / cos 8θ)/(sin 4θ / cos 4θ)

**Step 2: Simplify the left-hand side expression**

To simplify the left-hand side expression, we will rationalize the denominators.

Multiply the numerator and denominator of the first fraction by (cos 4θ)(cos 8θ) and the numerator and denominator of the second fraction by cos 4θ:

[(1/cos 8θ - 1)(cos 4θ)] / [(1/cos 4θ - 1)(cos 8θ)] = (sin 8θ / cos 8θ)/(sin 4θ / cos 4θ)

Now, distribute and simplify:

[(cos 4θ - cos 8θ) / (cos 8θ cos 4θ)] / [(cos 4θ - cos 8θ) / (cos 4θ cos 8θ)] = (sin 8θ / cos 8θ)/(sin 4θ / cos 4θ)

**Step 3: Cancel out common factors**

We can cancel out the common factors in the numerator and denominator:

[(cos 4θ - cos 8θ) / (cos 8θ cos 4θ)] * [(cos 4θ cos 8θ) / (cos 4θ - cos 8θ)] = (sin 8θ / cos 8θ)/(sin 4θ / cos 4θ)

(cos 4θ - cos 8θ) * (cos 4θ cos 8θ) = (sin 8θ / cos 8θ) * (cos 8θ - cos 4θ)

**Step 4: Use trigonometric identities**

We can apply the trigonometric identity cos A cos B = (1/2)[cos(A + B) + cos(A - B)] to simplify further:

[(cos 4θ cos 8θ) - (cos^2 8θ - cos^2 4θ)] = (sin 8θ / cos 8θ) * (cos 8θ - cos 4θ)

Now, use the identity cos^2 A = 1 - sin^2 A:

cos 4θ cos 8θ - (1 - sin^2 8θ - (1 - sin^2 4θ)) = (sin 8θ / cos 8θ) * (cos 8θ - cos 4θ)

cos 4θ cos 8θ - (1 - sin^2 8θ - 1 + sin^2