The value of the expression `log_10(tan6^@)+log_10(tan12^@)+log_10(tan18^@)+...+log_10(tan84^@)` is

Options

A. -1
B. 0
C. 1
D. 2

Explanation

To solve the expression \( \log_{10}(\tan 6^\circ) + \log_{10}(\tan 12^\circ) + \log_{10}(\tan 18^\circ) + \ldots + \log_{10}(\tan 84^\circ) \), we can follow these steps:

### Step 1: Use the logarithmic property
We can use the property of logarithms that states \( \log_a(m) + \log_a(n) = \log_a(m \cdot n) \). Therefore, we can combine all the logarithmic terms:

\[ E = \log_{10}(\tan 6^\circ \cdot \tan 12^\circ \cdot \tan 18^\circ \cdots \tan 84^\circ) \]

### Step 2: Identify pairs of angles
Notice that \( \tan(90^\circ - \theta) = \cot(\theta) \). We can pair the terms in the product:

- \( \tan 6^\circ \) pairs with \( \tan 84^\circ \)
- \( \tan 12^\circ \) pairs with \( \tan 78^\circ \)
- \( \tan 18^\circ \) pairs with \( \tan 72^\circ \)
- \( \tan 24^\circ \) pairs with \( \tan 66^\circ \)
- \( \tan 30^\circ \) pairs with \( \tan 60^\circ \)
- \( \tan 36^\circ \) pairs with \( \tan 54^\circ \)
- \( \tan 42^\circ \) pairs with \( \tan 48^\circ \)

### Step 3: Simplify the pairs
Using the identity \( \tan(90^\circ - x) = \cot(x) \), we can rewrite the pairs:

\[ \tan 6^\circ \cdot \tan 84^\circ = \tan 6^\circ \cdot \cot 6^\circ = 1 \]
\[ \tan 12^\circ \cdot \tan 78^\circ = \tan 12^\circ \cdot \cot 12^\circ = 1 \]
\[ \tan 18^\circ \cdot \tan 72^\circ = \tan 18^\circ \cdot \cot 18^\circ = 1 \]
\[ \tan 24^\circ \cdot \tan 66^\circ = \tan 24^\circ \cdot \cot 24^\circ = 1 \]
\[ \tan 30^\circ \cdot \tan 60^\circ = \tan 30^\circ \cdot \cot 30^\circ = 1 \]
\[ \tan 36^\circ \cdot \tan 54^\circ = \tan 36^\circ \cdot \cot 36^\circ = 1 \]
\[ \tan 42^\circ \cdot \tan 48^\circ = \tan 42^\circ \cdot \cot 42^\circ = 1 \]

### Step 4: Combine the results
Since each pair multiplies to 1, we have:

\[ \tan 6^\circ \cdot \tan 12^\circ \cdots \tan 84^\circ = 1 \cdot 1 \cdot 1 \cdots = 1 \]

### Step 5: Substitute back into the logarithm
Now substituting back into our expression for \( E \):

\[ E = \log_{10}(1) = 0 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{0} \]