The general solution of equation
`sin^2 theta sec theta + sqrt 3 tan theta = 0` is

Options

A. `theta = n pi+ (-1)^(n + 1) pi/3`
B. `theta = n pi`
C. `theta = n pi + (-1)^(n +1) pi/6`
D. `theta = (n pi)/2`

Answer

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

### Step 1: Rewrite the equation using trigonometric identities
We know that:
- \( \sec \theta = \frac{1}{\cos \theta} \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Substituting these identities into the equation gives:
\[ \sin^2 \theta \cdot \frac{1}{\cos \theta} + \sqrt{3} \cdot \frac{\sin \theta}{\cos \theta} = 0 \]

### Step 2: Simplify the equation
Multiply through by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \sin^2 \theta + \sqrt{3} \sin \theta = 0 \]

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

### Step 4: Set each factor to zero
Now, we can set each factor equal to zero:
1. \( \sin \theta = 0 \)
2. \( \sin \theta + \sqrt{3} = 0 \) which simplifies to \( \sin \theta = -\sqrt{3} \)

### Step 5: Solve for \( \theta \)
1. For \( \sin \theta = 0 \):
- The general solution is:
\[ \theta = n\pi, \quad n \in \mathbb{Z} \]

2. For \( \sin \theta = -\sqrt{3} \):
- Since the sine function only takes values between -1 and 1, \( \sin \theta = -\sqrt{3} \) has no solutions.

### Final Answer
Thus, the general solution of the equation is:
\[ \theta = n\pi, \quad n \in \mathbb{Z} \]

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