If `sin theta + cos theta = sqrt 3`, then what is `tan theta + cot theta` equal to ?

Options

A. 1
B. `sqrt2`
C. 2
D. `sqrt3`

Answer

To solve the problem, we need to find the value of \( \tan \theta + \cot \theta \) given that \( \sin \theta + \cos \theta = \sqrt{3} \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \sin \theta + \cos \theta = \sqrt{3} \]

2. **Square both sides:**
\[ (\sin \theta + \cos \theta)^2 = (\sqrt{3})^2 \]
This simplifies to:
\[ \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = 3 \]

3. **Use the Pythagorean identity:**
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute this into the equation:
\[ 1 + 2\sin \theta \cos \theta = 3 \]

4. **Isolate \( \sin \theta \cos \theta \):**
\[ 2\sin \theta \cos \theta = 3 - 1 \]
\[ 2\sin \theta \cos \theta = 2 \]
Divide both sides by 2:
\[ \sin \theta \cos \theta = 1 \]

5. **Express \( \tan \theta + \cot \theta \):**
Recall the definitions:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Therefore:
\[ \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]

6. **Substitute the known values:**
We already know \( \sin^2 \theta + \cos^2 \theta = 1 \) and \( \sin \theta \cos \theta = 1 \):
\[ \tan \theta + \cot \theta = \frac{1}{1} = 1 \]

### Final Answer:
\[ \tan \theta + \cot \theta = 1 \]