If `cos theta = (2x)/(1+x^2)` , find the values of `sin theta and cot theta` [Given `x^2 lt 1` ]

Answer

To solve the problem, we start with the given information and use trigonometric identities and the Pythagorean theorem.

### Step-by-Step Solution:

1. **Given Information**:
We are given that \( \cos \theta = \frac{2x}{1 + x^2} \) and \( x^2 < 1 \).

2. **Identify the Triangle**:
In a right triangle, we can represent \( \cos \theta \) as the ratio of the adjacent side (base) to the hypotenuse. Here, we can set:
- Base (adjacent side) = \( 2x \)
- Hypotenuse = \( 1 + x^2 \)

3. **Using Pythagorean Theorem**:
We need to find the length of the perpendicular side (opposite side). According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]
Plugging in the values:
\[ (1 + x^2)^2 = \text{Perpendicular}^2 + (2x)^2 \]

4. **Expand and Simplify**:
Expanding both sides:
\[ (1 + 2x^2 + x^4) = \text{Perpendicular}^2 + 4x^2 \]
Rearranging gives:
\[ \text{Perpendicular}^2 = (1 + 2x^2 + x^4) - 4x^2 \]
\[ \text{Perpendicular}^2 = 1 - 2x^2 + x^4 \]

5. **Recognize the Identity**:
The expression \( 1 - 2x^2 + x^4 \) can be factored as:
\[ \text{Perpendicular}^2 = (1 - x^2)^2 \]
Taking the square root gives:
\[ \text{Perpendicular} = 1 - x^2 \]

6. **Finding \( \sin \theta \)**:
We know that:
\[ \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{1 - x^2}{1 + x^2} \]

7. **Finding \( \cot \theta \)**:
We know that:
\[ \cot \theta = \frac{\text{Base}}{\text{Perpendicular}} = \frac{2x}{1 - x^2} \]

### Final Answers:
- \( \sin \theta = \frac{1 - x^2}{1 + x^2} \)
- \( \cot \theta = \frac{2x}{1 - x^2} \)