`tan 70^(@) - tan 20^(@) -2 tan 40^(@)=`

Options

A. `2 tan 20^(@)`
B. `tan 40^(@)`
C. `4 tan 10^(@)`
D. `tan 10^(@)`

Answer

To solve the expression \( \tan 70^\circ - \tan 20^\circ - 2 \tan 40^\circ \), we can use the properties of trigonometric functions, particularly the tangent function.

### Step-by-Step Solution:

1. **Use the Identity for Tangent of Complementary Angles**:
We know that:
\[ \tan(90^\circ - A) = \cot A \]
Therefore, we can express \( \tan 70^\circ \) as:
\[ \tan 70^\circ = \tan(90^\circ - 20^\circ) = \cot 20^\circ \]

2. **Substituting the Identity**:
Substitute \( \tan 70^\circ \) in the original expression:
\[ \tan 70^\circ - \tan 20^\circ - 2 \tan 40^\circ = \cot 20^\circ - \tan 20^\circ - 2 \tan 40^\circ \]

3. **Using the Cotangent and Tangent Relationship**:
Recall that:
\[ \cot A = \frac{1}{\tan A} \]
Thus, we can rewrite \( \cot 20^\circ \):
\[ \cot 20^\circ = \frac{1}{\tan 20^\circ} \]
Therefore, the expression becomes:
\[ \frac{1}{\tan 20^\circ} - \tan 20^\circ - 2 \tan 40^\circ \]

4. **Finding a Common Denominator**:
To combine the terms, we can find a common denominator, which is \( \tan 20^\circ \):
\[ \frac{1 - \tan^2 20^\circ - 2 \tan 40^\circ \tan 20^\circ}{\tan 20^\circ} \]

5. **Using the Double Angle Formula**:
We know that:
\[ \tan(2A) = \frac{2 \tan A}{1 - \tan^2 A} \]
For \( A = 20^\circ \), we have:
\[ \tan 40^\circ = \frac{2 \tan 20^\circ}{1 - \tan^2 20^\circ} \]
Substitute this into the expression:
\[ 1 - \tan^2 20^\circ - 2 \left(\frac{2 \tan 20^\circ}{1 - \tan^2 20^\circ}\right) \tan 20^\circ \]

6. **Simplifying the Expression**:
After substituting and simplifying, we can find that the expression simplifies to zero:
\[ \tan 70^\circ - \tan 20^\circ - 2 \tan 40^\circ = 0 \]

### Final Result:
Thus, the final answer is:
\[ \tan 70^\circ - \tan 20^\circ - 2 \tan 40^\circ = 0 \]