The value of `tan 4^@ tan 43^@ tan 47^@ tan 86^@` is

Options

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

Answer

To solve the expression \( \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ \), we can use some trigonometric identities and properties. Here’s the step-by-step solution:

### Step 1: Use the identity for tangent
Recall the identity:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
This means:
\[ \tan 86^\circ = \cot 4^\circ \quad \text{and} \quad \tan 47^\circ = \cot 43^\circ \]

### Step 2: Rewrite the expression
Using the identities from Step 1, we can rewrite the expression:
\[ \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ = \tan 4^\circ \tan 43^\circ \tan 47^\circ \cot 4^\circ \]

### Step 3: Simplify using cotangent
Now, we can simplify:
\[ \tan 4^\circ \cot 4^\circ = 1 \]
Thus, the expression simplifies to:
\[ 1 \cdot \tan 43^\circ \tan 47^\circ \]

### Step 4: Use another identity for tangent
Now, we can apply the identity again:
\[ \tan 43^\circ \tan 47^\circ = \tan 43^\circ \cot 43^\circ = 1 \]

### Step 5: Final result
Combining all the simplifications, we have:
\[ \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ = 1 \cdot 1 = 1 \]

### Conclusion
Thus, the value of \( \tan 4^\circ \tan 43^\circ \tan 47^\circ \tan 86^\circ \) is \( \boxed{1} \).

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