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

Options

A. `theta =npi+(-1)^(n+1)""(pi)/(3), theta =n pi , n in I`
B. `theta =n pi , n in I `
C. `theta =(n pi)/2 , n in I `
D. `theta =n pi +(-1)^(n+1)""(pi)/(3), n in 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
We start with the equation:
\[ \sin^2 \theta \sec \theta + \sqrt{3} \tan \theta = 0 \]
We can express \(\sec \theta\) 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 (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: Solve for \(\sin \theta\)
Now we have two cases to consider:
1. \(\sin \theta = 0\)
2. \(\sin \theta + \sqrt{3} = 0\)

For the first case:
\[ \sin \theta = 0 \implies \theta = n\pi \quad (n \in \mathbb{Z}) \]

For the second case:
\[ \sin \theta = -\sqrt{3} \]
However, since the sine function only takes values in the range \([-1, 1]\), \(-\sqrt{3}\) is not a valid solution.

### Step 5: Conclusion
Thus, the only solution comes from the first case:
\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]

### Final Answer
The general solution of the equation \( \sin^2 \theta \sec \theta + \sqrt{3} \tan \theta = 0 \) is:
\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]