The general solution of the equation
`"sin"^(2)x."sec"x+sqrt(3) tanx=0` is

Options

A. `(2pi+1)pi/(2)`
B. `npi`
C. `npi+pi/(6)`
D. `npi-pi/(6)`

Answer

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

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

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

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

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

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

### Step 5: Solve \( \sin x = 0 \)
The general solution for \( \sin x = 0 \) is:
\[ x = n\pi \quad \text{(where \( n \) is any integer)} \]

### Step 6: Solve \( \sin x + \sqrt{3} = 0 \)
This gives:
\[ \sin x = -\sqrt{3} \]
However, this is not possible since the range of \( \sin x \) is \([-1, 1]\). Therefore, we discard this case.

### Conclusion
The only valid solution comes from \( \sin x = 0 \):
\[ x = n\pi \]

Thus, the general solution of the equation \( \sin^2 x \sec x + \sqrt{3} \tan x = 0 \) is:
\[ \boxed{n\pi} \]