The value of `tan10^@tan50^@tan70^@` is

Options

A. `sqrt(3)`
B. `1/sqrt(3)`
C. 1
D. `-1`

Answer

To find the value of \( \tan 10^\circ \tan 50^\circ \tan 70^\circ \), we can use some trigonometric identities and properties.

### Step-by-Step Solution:

1. **Rewrite the angles**:
We know that \( \tan(70^\circ) = \cot(20^\circ) \) and \( \tan(50^\circ) = \cot(40^\circ) \). Therefore, we can express \( \tan(50^\circ) \) and \( \tan(70^\circ) \) in terms of \( \tan(10^\circ) \):
\[ \tan(70^\circ) = \frac{1}{\tan(20^\circ)} \]
\[ \tan(50^\circ) = \frac{1}{\tan(40^\circ)} \]

2. **Use the identity**:
We can use the identity \( \tan(90^\circ - x) = \cot(x) \) to rewrite:
\[ \tan(70^\circ) = \cot(20^\circ) = \frac{1}{\tan(20^\circ)} \]
Thus, we can express the product:
\[ \tan(10^\circ) \tan(50^\circ) \tan(70^\circ) = \tan(10^\circ) \tan(50^\circ) \cdot \frac{1}{\tan(20^\circ)} \]

3. **Combine the terms**:
Now, we can express \( \tan(50^\circ) \) in terms of \( \tan(10^\circ) \):
\[ \tan(50^\circ) = \tan(60^\circ - 10^\circ) = \frac{\tan(60^\circ) - \tan(10^\circ)}{1 + \tan(60^\circ) \tan(10^\circ)} \]
where \( \tan(60^\circ) = \sqrt{3} \).

4. **Substitute and simplify**:
Substitute \( \tan(60^\circ) \) into the equation:
\[ \tan(50^\circ) = \frac{\sqrt{3} - \tan(10^\circ)}{1 + \sqrt{3} \tan(10^\circ)} \]

5. **Final expression**:
Now, substituting \( \tan(50^\circ) \) back into the product:
\[ \tan(10^\circ) \cdot \frac{\sqrt{3} - \tan(10^\circ)}{1 + \sqrt{3} \tan(10^\circ)} \cdot \frac{1}{\tan(20^\circ)} \]

6. **Use the triple angle formula**:
We can use the formula \( \tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)} \) with \( \theta = 10^\circ \):
\[ \tan(30^\circ) = \frac{3\tan(10^\circ) - \tan^3(10^\circ)}{1 - 3\tan^2(10^\circ)} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we can equate and simplify to find the value.

7. **Final Calculation**:
After simplification, we find that:
\[ \tan(10^\circ) \tan(50^\circ) \tan(70^\circ) = \frac{1}{\sqrt{3}} \]

### Conclusion:
Thus, the value of \( \tan 10^\circ \tan 50^\circ \tan 70^\circ \) is \( \frac{1}{\sqrt{3}} \).