If 2 `sin(theta+(pi)/(3))=cos(theta-(pi)/(6))`, then `tantheta`=

Options

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

Explanation

To solve the equation \( 2 \sin\left(\theta + \frac{\pi}{3}\right) = \cos\left(\theta - \frac{\pi}{6}\right) \), we will follow these steps:

### Step 1: Expand both sides using angle addition formulas
Using the angle addition formulas:
- \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
- \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

We can expand the left side:
\[ 2 \sin\left(\theta + \frac{\pi}{3}\right) = 2 \left(\sin \theta \cos \frac{\pi}{3} + \cos \theta \sin \frac{\pi}{3}\right) \]
Substituting values:
\[ \cos \frac{\pi}{3} = \frac{1}{2}, \quad \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \]
Thus, the left side becomes:
\[ 2 \left(\sin \theta \cdot \frac{1}{2} + \cos \theta \cdot \frac{\sqrt{3}}{2}\right) = \sin \theta + \sqrt{3} \cos \theta \]

Now, expanding the right side:
\[ \cos\left(\theta - \frac{\pi}{6}\right) = \cos \theta \cos \frac{\pi}{6} + \sin \theta \sin \frac{\pi}{6} \]
Substituting values:
\[ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin \frac{\pi}{6} = \frac{1}{2} \]
Thus, the right side becomes:
\[ \cos \theta \cdot \frac{\sqrt{3}}{2} + \sin \theta \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \cos \theta + \frac{1}{2} \sin \theta \]

### Step 2: Set the expanded forms equal to each other
Now we have:
\[ \sin \theta + \sqrt{3} \cos \theta = \frac{\sqrt{3}}{2} \cos \theta + \frac{1}{2} \sin \theta \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ \sin \theta - \frac{1}{2} \sin \theta + \sqrt{3} \cos \theta - \frac{\sqrt{3}}{2} \cos \theta = 0 \]
This simplifies to:
\[ \frac{1}{2} \sin \theta + \left(\sqrt{3} - \frac{\sqrt{3}}{2}\right) \cos \theta = 0 \]
\[ \frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta = 0 \]

### Step 4: Isolate terms
Multiplying through by 2 to eliminate the fractions:
\[ \sin \theta + \sqrt{3} \cos \theta = 0 \]

### Step 5: Solve for \( \tan \theta \)
Rearranging gives:
\[ \sin \theta = -\sqrt{3} \cos \theta \]
Dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \tan \theta = -\sqrt{3} \]

### Final Result
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = -\sqrt{3} \]