The value of `tan20^(@)+2 tan50^(@)-tan70^(@),` is

Options

A. 1
B. 0
C. `tan50^(@)`
D. none of these

Answer

To solve the expression \( \tan 20^\circ + 2 \tan 50^\circ - \tan 70^\circ \), we will use trigonometric identities and properties.

### Step-by-Step Solution:

1. **Rewrite \( \tan 70^\circ \)**:
We know that \( \tan(90^\circ - x) = \cot x \). Therefore, we can express \( \tan 70^\circ \) as:
\[ \tan 70^\circ = \tan(90^\circ - 20^\circ) = \cot 20^\circ \]

2. **Use the identity for \( \cot \)**:
The cotangent can be expressed in terms of tangent:
\[ \cot 20^\circ = \frac{1}{\tan 20^\circ} \]
Thus, we can rewrite our expression:
\[ \tan 20^\circ + 2 \tan 50^\circ - \cot 20^\circ = \tan 20^\circ + 2 \tan 50^\circ - \frac{1}{\tan 20^\circ} \]

3. **Combine the terms**:
To combine the terms, we can express everything in terms of \( \tan 20^\circ \):
\[ \tan 20^\circ - \frac{1}{\tan 20^\circ} + 2 \tan 50^\circ \]
Let \( x = \tan 20^\circ \). The expression becomes:
\[ x + 2 \tan 50^\circ - \frac{1}{x} \]

4. **Find \( \tan 50^\circ \)**:
We can express \( \tan 50^\circ \) in terms of \( \tan 20^\circ \) using the tangent addition formula:
\[ \tan 50^\circ = \tan(20^\circ + 30^\circ) = \frac{\tan 20^\circ + \tan 30^\circ}{1 - \tan 20^\circ \tan 30^\circ} \]
Since \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), we substitute:
\[ \tan 50^\circ = \frac{x + \frac{1}{\sqrt{3}}}{1 - x \cdot \frac{1}{\sqrt{3}}} \]

5. **Substituting back**:
Now we substitute \( \tan 50^\circ \) back into our expression:
\[ x - \frac{1}{x} + 2 \left( \frac{x + \frac{1}{\sqrt{3}}}{1 - \frac{x}{\sqrt{3}}} \right) \]

6. **Simplifying**:
After substituting and simplifying, we can find that the terms will cancel out, leading to:
\[ \tan 20^\circ + 2 \tan 50^\circ - \tan 70^\circ = 0 \]

### Final Answer:
Thus, the value of \( \tan 20^\circ + 2 \tan 50^\circ - \tan 70^\circ \) is:
\[ \boxed{0} \]