The value of `tan((5pi)/12)-tan(pi/12)` is

Options

A. `sqrt(3)`
B. `2sqrt(3)`
C. `sqrt(3)/2`
D. `-sqrt(3)/2`

Answer

To find the value of \( \tan\left(\frac{5\pi}{12}\right) - \tan\left(\frac{\pi}{12}\right) \), we can follow these steps:

### Step 1: Rewrite \( \tan\left(\frac{5\pi}{12}\right) \)
Using the identity \( \tan\left(\theta\right) = \cot\left(\frac{\pi}{2} - \theta\right) \), we can express \( \tan\left(\frac{5\pi}{12}\right) \) as:
\[ \tan\left(\frac{5\pi}{12}\right) = \cot\left(\frac{\pi}{2} - \frac{5\pi}{12}\right) = \cot\left(\frac{6\pi}{12} - \frac{5\pi}{12}\right) = \cot\left(\frac{\pi}{12}\right) \]

### Step 2: Substitute into the expression
Now we can substitute this back into our original expression:
\[ \tan\left(\frac{5\pi}{12}\right) - \tan\left(\frac{\pi}{12}\right) = \cot\left(\frac{\pi}{12}\right) - \tan\left(\frac{\pi}{12}\right) \]

### Step 3: Rewrite \( \cot\left(\frac{\pi}{12}\right) \) and \( \tan\left(\frac{\pi}{12}\right) \)
We know that:
\[ \cot\left(\frac{\pi}{12}\right) = \frac{\cos\left(\frac{\pi}{12}\right)}{\sin\left(\frac{\pi}{12}\right)} \quad \text{and} \quad \tan\left(\frac{\pi}{12}\right) = \frac{\sin\left(\frac{\pi}{12}\right)}{\cos\left(\frac{\pi}{12}\right)} \]

### Step 4: Combine the fractions
Now we can combine these fractions:
\[ \cot\left(\frac{\pi}{12}\right) - \tan\left(\frac{\pi}{12}\right) = \frac{\cos\left(\frac{\pi}{12}\right)}{\sin\left(\frac{\pi}{12}\right)} - \frac{\sin\left(\frac{\pi}{12}\right)}{\cos\left(\frac{\pi}{12}\right)} \]
Finding a common denominator gives us:
\[ = \frac{\cos^2\left(\frac{\pi}{12}\right) - \sin^2\left(\frac{\pi}{12}\right)}{\sin\left(\frac{\pi}{12}\right) \cos\left(\frac{\pi}{12}\right)} \]

### Step 5: Use double angle identities
Using the double angle identities:
\[ \cos^2\theta - \sin^2\theta = \cos(2\theta) \quad \text{and} \quad \sin(2\theta) = 2\sin\theta\cos\theta \]
we can rewrite the expression as:
\[ = \frac{\cos\left(\frac{\pi}{6}\right)}{\frac{1}{2}\sin\left(\frac{\pi}{6}\right)} = \frac{2\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)} \]

### Step 6: Substitute known values
We know:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Substituting these values gives:
\[ = \frac{2 \cdot \frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{\frac{1}{2}} = 2\sqrt{3} \]

### Final Answer
Thus, the value of \( \tan\left(\frac{5\pi}{12}\right) - \tan\left(\frac{\pi}{12}\right) \) is:
\[ \boxed{2\sqrt{3}} \]