**Proof:**

Let's begin by using the trigonometric identity:

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

**Step 1: Simplifying tan(20 + 80):**

Using the identity above, we can simplify tan(20 + 80) as follows:

tan(20 + 80) = (tan 20 + tan 80) / (1 - tan 20 tan 80)

**Step 2: Simplifying tan(20 - 10):**

Now, let's consider another trigonometric identity:

tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

We can rewrite tan(20 + 80) as tan(20 - 10) using this identity:

tan(20 + 80) = tan(20 - 10) = (tan 20 - tan 10) / (1 + tan 20 tan 10)

**Step 3: Simplifying tan(20 - 10):**

Next, we can express tan(20 - 10) as follows:

tan(20 - 10) = tan(30)

Since tan(30) = 1/√3, we can substitute this value into our equation:

(tan 20 - tan 10) / (1 + tan 20 tan 10) = 1/√3

**Step 4: Simplifying tan(20 + 80):**

Now, let's go back to the equation from Step 1 and substitute the value we found in Step 3:

(tan 20 + tan 80) / (1 - tan 20 tan 80) = 1/√3

**Step 5: Cross multiplication:**

To simplify further, we can cross multiply the equation:

√3(tan 20 + tan 80) = 1 - tan 20 tan 80

**Step 6: Rearranging terms:**

Rearranging the terms, we get:

√3(tan 20 + tan 80) + tan 20 tan 80 - 1 = 0

**Step 7: Factoring:**

Now, we can factor the equation:

(√3 + tan 20 tan 80)(tan 20 + tan 80) - 1 = 0

**Step 8: Simplifying:**

We know that tan 20 + tan 80 = √3 (from the question statement). Substituting this into our equation, we get:

(√3 + tan 20 tan 80)(√3) - 1 = 0

**Step 9: Distributing and simplifying:**

Expanding the equation, we have:

3 + √3 tan 20 tan 80 - 1 = 0

Simplifying further, we get:

√3 tan 20 tan 80 + 2 = 0

**Step 10: Final result:**

Finally, we subtract 2 from both sides of the equation:

√3 tan 20 tan 80 = -2

Thus, we have proven that tan(20) tan(80) = √3 tan(50).