`tan9^@xxtan27^@xxtan63^@xxtan81^@=`
(a)4
(b)3
(c)2
(d)1

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the problem \( \tan 9^\circ \times \tan 27^\circ \times \tan 63^\circ \times \tan 81^\circ \), we can use the properties of trigonometric functions. Here’s the step-by-step solution:

### Step 1: Identify complementary angles
We know that:
- \( \tan(90^\circ - \theta) = \cot(\theta) \)

Using this property, we can rewrite:
- \( \tan 81^\circ = \cot 9^\circ \)
- \( \tan 63^\circ = \cot 27^\circ \)

### Step 2: Rewrite the expression
Now we can rewrite the original expression using the complementary angles:
\[ \tan 9^\circ \times \tan 27^\circ \times \tan 63^\circ \times \tan 81^\circ = \tan 9^\circ \times \tan 27^\circ \times \cot 27^\circ \times \cot 9^\circ \]

### Step 3: Simplify the expression
Using the identity \( \tan \theta \times \cot \theta = 1 \):
\[ \tan 9^\circ \times \cot 9^\circ = 1 \]
\[ \tan 27^\circ \times \cot 27^\circ = 1 \]

Thus, we can simplify:
\[ \tan 9^\circ \times \tan 27^\circ \times \cot 27^\circ \times \cot 9^\circ = 1 \times 1 = 1 \]

### Conclusion
Therefore, the value of \( \tan 9^\circ \times \tan 27^\circ \times \tan 63^\circ \times \tan 81^\circ \) is \( 1 \).

The answer is \( \boxed{1} \).

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