The general solution of the equation `sin^2thetasectheta+sqrt3 tantheta=0` is

Options

A. `theta = n pi + (-1)^(n+1) (pi)/(3), theta = n pi`
B. `theta = n pi`
C. `theta = npi + (-1)^(n+1) (pi)/(3)`
D. `theta = (n pi)/(2)`

Answer

To solve the equation \( \sin^2 \theta \sec \theta + \sqrt{3} \tan \theta = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sin^2 \theta \sec \theta + \sqrt{3} \tan \theta = 0 \]
We can rewrite \( \sec \theta \) and \( \tan \theta \) in terms of sine and cosine:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation gives:
\[ \sin^2 \theta \cdot \frac{1}{\cos \theta} + \sqrt{3} \cdot \frac{\sin \theta}{\cos \theta} = 0 \]

### Step 2: Combine the terms
Now, we can combine the terms over a common denominator:
\[ \frac{\sin^2 \theta + \sqrt{3} \sin \theta}{\cos \theta} = 0 \]
For this fraction to be zero, the numerator must be zero (since the denominator cannot be zero):
\[ \sin^2 \theta + \sqrt{3} \sin \theta = 0 \]

### Step 3: Factor the equation
We can factor the equation:
\[ \sin \theta (\sin \theta + \sqrt{3}) = 0 \]

### Step 4: Solve for \(\sin \theta\)
Setting each factor to zero gives us two equations to solve:
1. \( \sin \theta = 0 \)
2. \( \sin \theta + \sqrt{3} = 0 \) which simplifies to \( \sin \theta = -\sqrt{3} \)

### Step 5: Find the general solutions
For the first equation \( \sin \theta = 0 \):
- The general solution is:
\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]

For the second equation \( \sin \theta = -\sqrt{3} \):
- The value \(-\sqrt{3}\) is outside the range of the sine function, which is \([-1, 1]\). Therefore, this equation has no solution.

### Conclusion
The only valid solution comes from the first equation:
\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]