$\tan {20}^{\circ }\tan {40}^{\circ }\tan {60}^{\circ }\tan {80}^{\circ }=$

To solve tan 20° × tan 40° × tan 60° × tan 80°, we simplify it step by step.

Step 1: Recall the Value of tan 60°

From trigonometric tables, we know:

tan 60° = √3.

Thus, the expression becomes:

tan 20° × tan 40° × √3 × tan 80°.

Step 2: Use the Identity for tan(90° - θ)

The property tan(90° - θ) = cot(θ) simplifies tan 80°:

tan 80° = cot 10°.

Thus, the expression becomes:

tan 20° × tan 40° × √3 × cot 10°.

Step 3: Express cot 10° as 1/tan 10°

Using the definition cot x = 1/tan x, we have:

tan 20° × tan 40° × √3 × (1/tan 10°).

Step 4: Group Terms for Simplification

The expression becomes:

√3 × (tan 20° × tan 40° / tan 10°).

Step 5: Use Product Identities for tan

Trigonometric identities or values may now simplify specific terms further, but by evaluating directly, we find the approximate numerical value of this product. Through calculation or trigonometric tables:

tan 20° × tan 40° × tan 60° × tan 80° = 3.

Final Answer: tan 20° × tan 40° × tan 60° × tan 80° = 3.