General solution of `cotx+cosecx=sqrt(3)` is

Options

A. `2npi+(pi)/(6), ninZ`
B. `2npi+(pi)/(3), ninZ`
C. `2npi+(pi)/(4), ninZ`
D. `2npi+(pi)/(2), ninZ`

Answer

To find the general solution of the equation \( \cot x + \csc x = \sqrt{3} \), we can follow these steps:

### Step 1: Rewrite the equation using sine and cosine
We start by expressing cotangent and cosecant in terms of sine and cosine:
\[ \cot x = \frac{\cos x}{\sin x}, \quad \csc x = \frac{1}{\sin x} \]
Thus, the equation becomes:
\[ \frac{\cos x}{\sin x} + \frac{1}{\sin x} = \sqrt{3} \]

### Step 2: Combine the fractions
Now, we can combine the fractions on the left-hand side:
\[ \frac{\cos x + 1}{\sin x} = \sqrt{3} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ \cos x + 1 = \sqrt{3} \sin x \]

### Step 4: Rearranging the equation
Rearranging this, we have:
\[ \sqrt{3} \sin x - \cos x - 1 = 0 \]

### Step 5: Square both sides
To eliminate the square root, we can square both sides:
\[ (\sqrt{3} \sin x - \cos x - 1)^2 = 0 \]
Expanding this gives:
\[ 3 \sin^2 x - 2\sqrt{3} \sin x \cos x - 2\sqrt{3} \sin x + \cos^2 x + 2\cos x + 1 = 0 \]

### Step 6: Use the Pythagorean identity
Using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ 3(1 - \cos^2 x) - 2\sqrt{3} \sin x \cos x - 2\sqrt{3} \sin x + \cos^2 x + 2\cos x + 1 = 0 \]
This simplifies to:
\[ (3 - 3\cos^2 x + \cos^2 x + 1) - 2\sqrt{3} \sin x \cos x - 2\sqrt{3} \sin x = 0 \]
\[ (4 - 2\cos^2 x) - 2\sqrt{3} \sin x (\cos x + 1) = 0 \]

### Step 7: Solve for \( \cos x \)
This leads us to two cases:
1. \( \cos x + 1 = 0 \)
2. \( 2 - 2\cos^2 x - 2\sqrt{3} \sin x = 0 \)

From the first case:
\[ \cos x = -1 \Rightarrow x = \pi + 2n\pi, \quad n \in \mathbb{Z} \]

From the second case, we can solve for \( \cos x = \frac{1}{2} \):
\[ \cos x = \frac{1}{2} \Rightarrow x = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad x = -\frac{\pi}{3} + 2n\pi \]

### Step 8: Combine the solutions
The general solutions are:
\[ x = \pi + 2n\pi \quad \text{and} \quad x = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad x = -\frac{\pi}{3} + 2n\pi \]

### Final General Solution
Thus, the general solution of the equation \( \cot x + \csc x = \sqrt{3} \) is:
\[ x = \pi + 2n\pi \quad \text{or} \quad x = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad x = -\frac{\pi}{3} + 2n\pi, \quad n \in \mathbb{Z} \]