Solve ` cot theta + "cosec" theta =sqrt(3)`

Answer

To solve the equation \( \cot \theta + \csc \theta = \sqrt{3} \), we can follow these steps:

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

### Step 2: Combine the fractions
Next, we combine the fractions on the left-hand side:
\[ \frac{\cos \theta + 1}{\sin \theta} = \sqrt{3} \]
Now, we can cross-multiply to eliminate the fraction:
\[ \cos \theta + 1 = \sqrt{3} \sin \theta \]

### Step 3: Square both sides
To eliminate the sine and cosine terms, we square both sides:
\[ (\cos \theta + 1)^2 = (\sqrt{3} \sin \theta)^2 \]
Expanding both sides gives:
\[ \cos^2 \theta + 2\cos \theta + 1 = 3\sin^2 \theta \]

### Step 4: Use the Pythagorean identity
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \), so we can express \( \sin^2 \theta \) as \( 1 - \cos^2 \theta \):
\[ \cos^2 \theta + 2\cos \theta + 1 = 3(1 - \cos^2 \theta) \]
Simplifying this gives:
\[ \cos^2 \theta + 2\cos \theta + 1 = 3 - 3\cos^2 \theta \]
Combining like terms results in:
\[ 4\cos^2 \theta + 2\cos \theta - 2 = 0 \]

### Step 5: Simplify the quadratic equation
Dividing the entire equation by 2:
\[ 2\cos^2 \theta + \cos \theta - 1 = 0 \]

### Step 6: Factor the quadratic equation
Now we can factor the quadratic:
\[ (2\cos \theta + 2)(\cos \theta - 1) = 0 \]
Setting each factor to zero gives:
1. \( 2\cos \theta + 2 = 0 \) → \( \cos \theta = -1 \)
2. \( \cos \theta - 1 = 0 \) → \( \cos \theta = \frac{1}{2} \)

### Step 7: Solve for \( \theta \)
For \( \cos \theta = -1 \):
\[ \theta = \pi + 2n\pi, \quad n \in \mathbb{Z} \]
For \( \cos \theta = \frac{1}{2} \):
\[ \theta = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \theta = \frac{5\pi}{3} + 2n\pi, \quad n \in \mathbb{Z} \]

### Final Answer
Thus, the solutions for \( \theta \) are:
\[ \theta = \pi + 2n\pi, \quad \theta = \frac{\pi}{3} + 2n\pi, \quad \text{or} \quad \theta = \frac{5\pi}{3} + 2n\pi, \quad n \in \mathbb{Z} \]