If `cottheta=1/p`, then `sin2theta` is equal to

Options

A. `1/(1+p^2)`
B. `(2p)/(1+p^2)`
C. `p^2/(1+p)`
D. `(1+p^2)/(1+p)`

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 have \( \cot \theta = \frac{1}{p} \).

2. **Understanding Cotangent**: Recall that \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \). Therefore, we can set:
- Adjacent (base) = 1
- Opposite (perpendicular) = p

3. **Finding the Hypotenuse**: We apply the Pythagorean theorem to find the hypotenuse (\( h \)):
\[ h = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{p^2 + 1^2} = \sqrt{p^2 + 1} \]

4. **Finding Sine and Cosine**:
- \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{p}{\sqrt{p^2 + 1}} \)
- \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{p^2 + 1}} \)

5. **Using the Double Angle Formula**: We know that:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Substituting the values we found:
\[ \sin 2\theta = 2 \left(\frac{p}{\sqrt{p^2 + 1}}\right) \left(\frac{1}{\sqrt{p^2 + 1}}\right) \]

6. **Simplifying the Expression**:
\[ \sin 2\theta = 2 \cdot \frac{p \cdot 1}{p^2 + 1} = \frac{2p}{p^2 + 1} \]

7. **Final Answer**: Therefore, we conclude that:
\[ \sin 2\theta = \frac{2p}{p^2 + 1} \]