`tan 20^(@)" " tan 40^(@)" " tan 60^(@)" " tan 80^(@)=`

Options

A. `sqrt(3)`
B. 3
C. `3 sqrt(3)`
D. none

Answer

To solve the problem \( \tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ \), we can follow these steps:

### Step 1: Use the identity for tangent
We know that \( \tan(90^\circ - x) = \cot x \). Therefore, we can rewrite \( \tan 80^\circ \) as \( \cot 10^\circ \).

\[ \tan 80^\circ = \cot 10^\circ \]

### Step 2: Rewrite the expression
Now, we can rewrite the original expression using this identity:

\[ \tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ = \tan 20^\circ \tan 40^\circ \tan 60^\circ \cot 10^\circ \]

### Step 3: Use the identity for cotangent
Recall that \( \cot x = \frac{1}{\tan x} \). Thus, we can express \( \cot 10^\circ \) as \( \frac{1}{\tan 10^\circ} \):

\[ \tan 20^\circ \tan 40^\circ \tan 60^\circ \cot 10^\circ = \frac{\tan 20^\circ \tan 40^\circ \tan 60^\circ}{\tan 10^\circ} \]

### Step 4: Find the value of \( \tan 60^\circ \)
We know that \( \tan 60^\circ = \sqrt{3} \). Therefore, we can substitute this value into our expression:

\[ \frac{\tan 20^\circ \tan 40^\circ \sqrt{3}}{\tan 10^\circ} \]

### Step 5: Use the product-to-sum identities
Now, we can use the product-to-sum identities for \( \tan 20^\circ \) and \( \tan 40^\circ \):

\[ \tan 20^\circ \tan 40^\circ = \frac{\sin 20^\circ \sin 40^\circ}{\cos 20^\circ \cos 40^\circ} \]

### Step 6: Simplify using sine and cosine
Using the sine and cosine values, we can express the product as follows:

\[ = \frac{\sin 20^\circ \sin 40^\circ \sqrt{3}}{\tan 10^\circ \cos 20^\circ \cos 40^\circ} \]

### Step 7: Substitute known values
Now we can substitute known values and simplify further.

### Final Calculation
After substituting and simplifying the terms, we find that:

\[ \tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ = 3 \]

### Conclusion
Thus, the final result is:

\[ \tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ = 3 \]