If `y=sqrt(((1+cos2theta)/(1-cos2theta))),(dy)/(d theta)` at `theta=(3pi)/(4)` is

Options

A. `-2`
B. `2`
C. `+-2`
D. None of these

Answer

To solve the problem, we need to find the derivative \( \frac{dy}{d\theta} \) of the function \( y = \sqrt{\frac{1 + \cos 2\theta}{1 - \cos 2\theta}} \) at \( \theta = \frac{3\pi}{4} \).

### Step-by-Step Solution:

1. **Rewrite the Function**:
We start with the function:
\[ y = \sqrt{\frac{1 + \cos 2\theta}{1 - \cos 2\theta}} \]

2. **Use Trigonometric Identities**:
We can simplify \( 1 + \cos 2\theta \) and \( 1 - \cos 2\theta \) using the double angle identities:
\[ 1 + \cos 2\theta = 2\cos^2 \theta \quad \text{and} \quad 1 - \cos 2\theta = 2\sin^2 \theta \]
Therefore, we can rewrite \( y \):
\[ y = \sqrt{\frac{2\cos^2 \theta}{2\sin^2 \theta}} = \sqrt{\frac{\cos^2 \theta}{\sin^2 \theta}} = \frac{|\cos \theta|}{|\sin \theta|} = |\cot \theta| \]

3. **Determine the Sign of \( \cot \theta \)**:
The value of \( y \) depends on the quadrant in which \( \theta \) lies:
- In the first and third quadrants, \( \cot \theta \) is positive, so \( y = \cot \theta \).
- In the second and fourth quadrants, \( \cot \theta \) is negative, so \( y = -\cot \theta \).

4. **Differentiate \( y \)**:
We differentiate \( y \) with respect to \( \theta \):
- For \( \theta \) in the first and third quadrants:
\[ \frac{dy}{d\theta} = -\csc^2 \theta \]
- For \( \theta \) in the second and fourth quadrants:
\[ \frac{dy}{d\theta} = \csc^2 \theta \]

5. **Evaluate \( \frac{dy}{d\theta} \) at \( \theta = \frac{3\pi}{4} \)**:
The angle \( \frac{3\pi}{4} \) is in the second quadrant, where \( y = -\cot \theta \). Thus, we use:
\[ \frac{dy}{d\theta} = \csc^2 \theta \]
We need to calculate \( \csc^2 \left(\frac{3\pi}{4}\right) \):
\[ \sin \left(\frac{3\pi}{4}\right) = \sin \left(\pi - \frac{\pi}{4}\right) = \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]
Therefore:
\[ \csc \left(\frac{3\pi}{4}\right) = \frac{1}{\sin \left(\frac{3\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]
Thus:
\[ \csc^2 \left(\frac{3\pi}{4}\right) = (\sqrt{2})^2 = 2 \]

6. **Final Result**:
Therefore, the value of \( \frac{dy}{d\theta} \) at \( \theta = \frac{3\pi}{4} \) is:
\[ \frac{dy}{d\theta} \bigg|_{\theta = \frac{3\pi}{4}} = 2 \]