Evaluate: tan 20° × tan 40° × tan 60° × tan 80°

To solve this trigonometry problem, we need to understand how tangent functions work and how we can simplify them using identities.

First, recall an important identity: 
tan θ = sin θ / cos θ

Using this, we can rewrite the expression as:

(sin 20° × sin 40° × sin 60° × sin 80°) / (cos 20° × cos 40° × cos 60° × cos 80°)

Now, we simplify step by step. We already know that: 
tan 60° = √3

So we can separate it:

tan 20° × tan 40° × tan 80° × √3

There is a known trigonometric result: 
tan 20° × tan 40° × tan 80° = √3

Now multiply: 
√3 × √3 = 3

So, the final value becomes: 
tan 20° × tan 40° × tan 60° × tan 80° = 3

Final Answer:

3

Tip for Students:
Whenever you see products of tangent with angles like 20°, 40°, 60°, 80°, try converting into sine and cosine or use known identities. This makes solving much easier!