Express `sin theta ` in terms of ` cot theta`.

Answer

To express \( \sin \theta \) in terms of \( \cot \theta \), we can follow these steps:

### Step 1: Recall the Pythagorean Identity
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

### Step 2: Express \( \cos^2 \theta \) in terms of \( \cot \theta \)
We also know the relationship involving cotangent:
\[ \cot^2 \theta + 1 = \csc^2 \theta \]
This can be rearranged to express \( \cos^2 \theta \):
\[ \cos^2 \theta = \frac{1}{\csc^2 \theta} = \frac{1}{1 + \cot^2 \theta} \]

### Step 3: Substitute \( \cos^2 \theta \) into the Pythagorean Identity
Now, we can substitute \( \cos^2 \theta \) into the Pythagorean identity:
\[ \sin^2 \theta + \frac{1}{1 + \cot^2 \theta} = 1 \]

### Step 4: Solve for \( \sin^2 \theta \)
Rearranging gives us:
\[ \sin^2 \theta = 1 - \frac{1}{1 + \cot^2 \theta} \]
To combine the terms, we can find a common denominator:
\[ \sin^2 \theta = \frac{(1 + \cot^2 \theta) - 1}{1 + \cot^2 \theta} = \frac{\cot^2 \theta}{1 + \cot^2 \theta} \]

### Step 5: Take the square root to find \( \sin \theta \)
Now, we take the square root of both sides:
\[ \sin \theta = \sqrt{\frac{\cot^2 \theta}{1 + \cot^2 \theta}} = \frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}} \]

### Final Expression
Thus, we have expressed \( \sin \theta \) in terms of \( \cot \theta \):
\[ \sin \theta = \frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}} \]