For `n in Z ,` the general solution of `(sqrt(3)-1)sintheta+(sqrt(3)+1)costheta=2i s(n in Z)` `theta=2npi+-pi/4+pi/(12)` `theta=npi+(-1)^npi/4+pi/(12)` `theta=2npi+-pi/4` `theta=npi+(-1)^npi/4-pi/(12)`

Options

A. `theta=2npi pm pi/4+pi/12`
B. `theta=n pi +(-1)^(n) pi/4+pi/12`
C. `theta=2npi pm pi/4`
D. `theta=npi +(-1)^(n) pi/4- pi/12`

Explanation

To solve the equation \((\sqrt{3}-1)\sin\theta + (\sqrt{3}+1)\cos\theta = 2\) for \(\theta\) in terms of \(n \in \mathbb{Z}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (\sqrt{3}-1)\sin\theta + (\sqrt{3}+1)\cos\theta = 2 \]

### Step 2: Normalize the equation
To simplify, we can divide the entire equation by \(2\sqrt{2}\):
\[ \frac{\sqrt{3}-1}{2\sqrt{2}}\sin\theta + \frac{\sqrt{3}+1}{2\sqrt{2}}\cos\theta = \frac{2}{2\sqrt{2}} \]
This simplifies to:
\[ \frac{\sqrt{3}-1}{2\sqrt{2}}\sin\theta + \frac{\sqrt{3}+1}{2\sqrt{2}}\cos\theta = \frac{1}{\sqrt{2}} \]

### Step 3: Identify sine and cosine values
We recognize that:
\[ \sin\left(\frac{\pi}{12}\right) = \frac{\sqrt{3}-1}{2\sqrt{2}} \quad \text{and} \quad \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{3}+1}{2\sqrt{2}} \]
Thus, we can rewrite the equation as:
\[ \sin\left(\frac{\pi}{12}\right)\sin\theta + \cos\left(\frac{\pi}{12}\right)\cos\theta = \frac{1}{\sqrt{2}} \]

### Step 4: Use the cosine angle subtraction identity
Using the cosine angle subtraction identity:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
we can express our equation as:
\[ \cos\left(\theta - \frac{\pi}{12}\right) = \frac{1}{\sqrt{2}} \]

### Step 5: Solve for \(\theta\)
The general solution for \(\cos x = k\) is given by:
\[ x = 2n\pi \pm \alpha \]
where \(\alpha = \frac{\pi}{4}\) (since \(\cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}\)). Therefore, we have:
\[ \theta - \frac{\pi}{12} = 2n\pi \pm \frac{\pi}{4} \]
This leads to:
\[ \theta = 2n\pi \pm \frac{\pi}{4} + \frac{\pi}{12} \]

### Step 6: Combine the terms
To combine the terms, we need a common denominator:
\[ \frac{\pi}{4} = \frac{3\pi}{12} \]
Thus, we can rewrite:
\[ \theta = 2n\pi \pm \left(\frac{3\pi}{12} + \frac{\pi}{12}\right) = 2n\pi \pm \frac{4\pi}{12} = 2n\pi \pm \frac{\pi}{3} \]

### Final General Solution
Thus, the general solution for \(\theta\) is:
\[ \theta = 2n\pi \pm \frac{\pi}{4} + \frac{\pi}{12} \]

### Conclusion
The correct answer is:
\[ \theta = 2n\pi \pm \frac{\pi}{4} + \frac{\pi}{12} \]