`tan 20^(@)+ tan 40^(@) + tan 60^(@) +"…….."+tan 160^(@) + tan 180^(@)=`

Options

A. 1
B. 2
C. 0
D. none

Answer

To solve the equation \( \tan 20^\circ + \tan 40^\circ + \tan 60^\circ + \ldots + \tan 160^\circ + \tan 180^\circ \), we can use the properties of the tangent function.

### Step-by-Step Solution:

1. **Identify the Terms**:
The series consists of the tangent values from \( 20^\circ \) to \( 180^\circ \) in increments of \( 20^\circ \):
\[ \tan 20^\circ, \tan 40^\circ, \tan 60^\circ, \tan 80^\circ, \tan 100^\circ, \tan 120^\circ, \tan 140^\circ, \tan 160^\circ, \tan 180^\circ \]

2. **Use the Property of Tangent**:
We know that:
\[ \tan(180^\circ - \theta) = -\tan(\theta) \]
This property allows us to pair terms in the series:
- \( \tan 20^\circ \) pairs with \( \tan 160^\circ \)
- \( \tan 40^\circ \) pairs with \( \tan 140^\circ \)
- \( \tan 60^\circ \) pairs with \( \tan 120^\circ \)
- \( \tan 80^\circ \) pairs with \( \tan 100^\circ \)

3. **Calculate the Pairs**:
Each pair sums to zero:
- \( \tan 20^\circ + \tan 160^\circ = \tan 20^\circ - \tan 20^\circ = 0 \)
- \( \tan 40^\circ + \tan 140^\circ = \tan 40^\circ - \tan 40^\circ = 0 \)
- \( \tan 60^\circ + \tan 120^\circ = \tan 60^\circ - \tan 60^\circ = 0 \)
- \( \tan 80^\circ + \tan 100^\circ = \tan 80^\circ - \tan 80^\circ = 0 \)

4. **Include the Last Term**:
The last term is \( \tan 180^\circ \), which is equal to \( 0 \).

5. **Sum All the Values**:
Adding all these pairs and the last term gives:
\[ 0 + 0 + 0 + 0 + 0 = 0 \]

### Final Answer:
Thus, the value of the entire expression is:
\[ \boxed{0} \]