What is the value of ` tan 20^(@) tan 60^(@) tan 70^(@)` ?

Options

A. 3
B. `sqrt""3`
C. 1
D. None of these

Answer

To find the value of \( \tan 20^\circ \tan 60^\circ \tan 70^\circ \), we can use some trigonometric identities and properties. Let's break down the solution step by step.

### Step 1: Use the identity for tangent
We know that:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
Using this identity, we can express \( \tan 20^\circ \) in terms of \( \tan 70^\circ \):
\[ \tan 20^\circ = \tan(90^\circ - 70^\circ) = \cot 70^\circ \]

### Step 2: Substitute the identity into the expression
Now we can rewrite the original expression:
\[ \tan 20^\circ \tan 60^\circ \tan 70^\circ = \cot 70^\circ \tan 60^\circ \tan 70^\circ \]

### Step 3: Simplify the expression
Recall that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we can substitute \( \cot 70^\circ \) as follows:
\[ \cot 70^\circ = \frac{1}{\tan 70^\circ} \]
So, the expression becomes:
\[ \frac{1}{\tan 70^\circ} \tan 60^\circ \tan 70^\circ \]

### Step 4: Cancel out \( \tan 70^\circ \)
Now we can cancel \( \tan 70^\circ \):
\[ = \tan 60^\circ \]

### Step 5: Find the value of \( \tan 60^\circ \)
We know from trigonometric values that:
\[ \tan 60^\circ = \sqrt{3} \]

### Final Answer
Thus, the value of \( \tan 20^\circ \tan 60^\circ \tan 70^\circ \) is:
\[ \sqrt{3} \]