The general solution of `sin^2 theta sec theta +sqrt3 tan theta=0` is `(n in 1)`

Options

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

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 using trigonometric identities
We know that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, we can rewrite the equation as:

\[ \sin^2 \theta \cdot \frac{1}{\cos \theta} + \sqrt{3} \cdot \frac{\sin \theta}{\cos \theta} = 0 \]

### Step 2: Combine the terms over a common denominator
Multiplying through by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)) gives:

\[ \sin^2 \theta + \sqrt{3} \sin \theta = 0 \]

### 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
This 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: Solve \( \sin \theta = 0 \)
The general solution for \( \sin \theta = 0 \) is:

\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]

### Step 6: Solve \( \sin \theta = -\sqrt{3} \)
However, \( \sin \theta = -\sqrt{3} \) is not possible since the sine function only takes values in the range \([-1, 1]\). Therefore, this factor does not provide any valid solutions.

### Conclusion
The only valid solution is from the first factor:

\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]

Thus, the general solution of the equation is:

\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]