Find the value of `tan8^(@)tan22^(@)cot60^(@)tan68^(@)tan82^(@)`

Options

A. `sqrt(3)`
B. 1
C. `(2)/(sqrt(3))`
D. `(1)/(sqrt(3))`

Answer

To find the value of \( \tan 8^\circ \tan 22^\circ \cot 60^\circ \tan 68^\circ \tan 82^\circ \), we can use the properties of trigonometric functions.

### Step 1: Rewrite cotangent
We know that:
\[ \cot 60^\circ = \frac{1}{\tan 60^\circ} \]
Since \( \tan 60^\circ = \sqrt{3} \), we have:
\[ \cot 60^\circ = \frac{1}{\sqrt{3}} \]

### Step 2: Use the complementary angle identities
Using the identity \( \tan(90^\circ - \theta) = \cot \theta \), we can rewrite some of the terms:
\[ \tan 68^\circ = \cot 22^\circ \quad \text{and} \quad \tan 82^\circ = \cot 8^\circ \]

### Step 3: Substitute the values
Now we can substitute these identities into our expression:
\[ \tan 8^\circ \tan 22^\circ \cot 60^\circ \tan 68^\circ \tan 82^\circ = \tan 8^\circ \tan 22^\circ \cdot \frac{1}{\sqrt{3}} \cdot \cot 22^\circ \cdot \cot 8^\circ \]

### Step 4: Simplify the expression
Now, we can simplify:
\[ \tan 8^\circ \cot 8^\circ = 1 \quad \text{and} \quad \tan 22^\circ \cot 22^\circ = 1 \]
Thus, the expression simplifies to:
\[ 1 \cdot 1 \cdot \frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Final Answer
Therefore, the value of \( \tan 8^\circ \tan 22^\circ \cot 60^\circ \tan 68^\circ \tan 82^\circ \) is:
\[ \frac{1}{\sqrt{3}} \]

### Options
From the options provided, the correct answer is option D: \( \frac{1}{\sqrt{3}} \).

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