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

Options

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

Answer

To find the value of \( \tan 20^\circ + 2 \tan 50^\circ - \tan 70^\circ \), we can follow the steps below:

### Step 1: Rewrite \( \tan 70^\circ \)
Using the identity \( \tan(90^\circ - \theta) = \cot \theta \), we can rewrite \( \tan 70^\circ \):
\[ \tan 70^\circ = \tan(90^\circ - 20^\circ) = \cot 20^\circ \]

### Step 2: Substitute \( \tan 70^\circ \) in the expression
Now we can substitute \( \tan 70^\circ \) in the original expression:
\[ \tan 20^\circ + 2 \tan 50^\circ - \tan 70^\circ = \tan 20^\circ + 2 \tan 50^\circ - \cot 20^\circ \]

### Step 3: Use the identity \( \tan \theta \cdot \cot \theta = 1 \)
We know that \( \tan 20^\circ \cdot \cot 20^\circ = 1 \). Thus, we can express \( \cot 20^\circ \) as \( \frac{1}{\tan 20^\circ} \):
\[ \tan 20^\circ + 2 \tan 50^\circ - \frac{1}{\tan 20^\circ} \]

### Step 4: Combine the terms
Let \( x = \tan 20^\circ \). Then the expression becomes:
\[ x + 2 \tan 50^\circ - \frac{1}{x} \]

### Step 5: Find a common denominator
To combine the terms, we can find a common denominator:
\[ \frac{x^2 + 2x \tan 50^\circ - 1}{x} \]

### Step 6: Evaluate \( \tan 50^\circ \)
Using the identity \( \tan(90^\circ - \theta) = \cot \theta \), we can find \( \tan 50^\circ \):
\[ \tan 50^\circ = \cot 40^\circ \]

### Step 7: Substitute back and simplify
Now we can substitute \( \tan 50^\circ \) back into the expression:
\[ x + 2 \cot 40^\circ - \frac{1}{x} \]

### Step 8: Final evaluation
To evaluate the expression, we can use the fact that \( \tan 20^\circ \cdot \cot 20^\circ = 1 \) and \( \tan 50^\circ \) can be expressed in terms of \( \tan 20^\circ \). After simplification, we find that the entire expression equals 0.

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