Find the value of `tan80^(@)`

Options

A. `tan70^(@)+tan10^(@)`
B. `tan70^(@)+2tan10^(@)`
C. `2tan70^(@)+tan10^(@)`
D. `2tan70^(@)+2tan10^(@)`

Answer

To find the value of \( \tan 80^\circ \), we can use the tangent addition formula. The formula states that:

\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]

### Step-by-Step Solution:

1. **Identify the angles**: We can express \( 80^\circ \) as \( 70^\circ + 10^\circ \).

2. **Apply the tangent addition formula**:
\[ \tan(80^\circ) = \tan(70^\circ + 10^\circ) = \frac{\tan 70^\circ + \tan 10^\circ}{1 - \tan 70^\circ \tan 10^\circ} \]

3. **Use the identity**: We know that \( \tan(90^\circ - x) = \cot x \). Therefore, \( \tan 70^\circ = \cot 20^\circ \).

4. **Substituting values**:
\[ \tan 70^\circ = \cot 20^\circ = \frac{1}{\tan 20^\circ} \]
Thus, we can rewrite the equation:
\[ \tan(80^\circ) = \frac{\frac{1}{\tan 20^\circ} + \tan 10^\circ}{1 - \frac{1}{\tan 20^\circ} \tan 10^\circ} \]

5. **Simplifying the equation**:
- The numerator becomes:
\[ \frac{1 + \tan 10^\circ \tan 20^\circ}{\tan 20^\circ} \]
- The denominator becomes:
\[ 1 - \frac{\tan 10^\circ}{\tan 20^\circ} = \frac{\tan 20^\circ - \tan 10^\circ}{\tan 20^\circ} \]

6. **Combining the fractions**:
\[ \tan(80^\circ) = \frac{1 + \tan 10^\circ \tan 20^\circ}{\tan 20^\circ - \tan 10^\circ} \]

7. **Using known values**: If we know the values of \( \tan 10^\circ \) and \( \tan 20^\circ \), we can substitute them to find \( \tan 80^\circ \).

8. **Final calculation**: After substituting the values, we can calculate the final result.

### Final Answer:
The value of \( \tan 80^\circ \) can be approximated using a calculator or trigonometric tables, which gives \( \tan 80^\circ \approx 5.6713 \).