Let `(3pi)/4 < theta < pi` and `sqrt(2 cot theta+1/sin^2 theta) = k - cot theta` then `k=`

Options

A. 1
B. `-1`
C. 0
D. `1/2`

Explanation

To solve the equation \( \sqrt{2 \cot \theta + \frac{1}{\sin^2 \theta}} = k - \cot \theta \) given that \( \frac{3\pi}{4} < \theta < \pi \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the left-hand side of the equation:
\[ \sqrt{2 \cot \theta + \frac{1}{\sin^2 \theta}} \]
We know that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Thus, we can rewrite \( 2 \cot \theta \) as:
\[ 2 \cot \theta = \frac{2 \cos \theta}{\sin \theta} \]
Now, substituting this into the expression gives:
\[ \sqrt{\frac{2 \cos \theta}{\sin \theta} + \frac{1}{\sin^2 \theta}} \]

### Step 2: Combine the terms under a common denominator
The expression can be combined under a common denominator:
\[ \sqrt{\frac{2 \cos \theta \sin + 1}{\sin^2 \theta}} \]
This simplifies to:
\[ \sqrt{\frac{2 \cos \theta \sin + 1}{\sin^2 \theta}} = \frac{\sqrt{2 \cos \theta \sin + 1}}{\sin \theta} \]

### Step 3: Use the Pythagorean identity
Recall the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Thus, we can express \( 1 \) as \( \sin^2 \theta + \cos^2 \theta \):
\[ \sqrt{2 \cos \theta \sin + \sin^2 \theta + \cos^2 \theta} = \sqrt{(\sin \theta + \cos \theta)^2} \]
This simplifies to:
\[ \frac{|\sin \theta + \cos \theta|}{\sin \theta} \]

### Step 4: Determine the sign of the expression
Since \( \frac{3\pi}{4} < \theta < \pi \), we are in the second quadrant where both \( \sin \theta \) is positive and \( \cos \theta \) is negative. Therefore, \( \sin \theta + \cos \theta \) will also be positive. Thus:
\[ |\sin \theta + \cos \theta| = \sin \theta + \cos \theta \]
So we have:
\[ \frac{\sin \theta + \cos \theta}{\sin \theta} = 1 + \frac{\cos \theta}{\sin \theta} = 1 + \cot \theta \]

### Step 5: Set the equation
Now we can equate this to the right-hand side of the original equation:
\[ 1 + \cot \theta = k - \cot \theta \]

### Step 6: Solve for \( k \)
Rearranging gives:
\[ k = 1 + 2\cot \theta \]

### Step 7: Determine \( k \)
In the second quadrant, \( \cot \theta \) is negative. However, we need to find \( k \) independent of the specific value of \( \cot \theta \). Since we are looking for a constant \( k \) that satisfies the equation for all \( \theta \) in the given range, we can evaluate at any specific point. For example, if we consider \( \theta = \frac{3\pi}{4} \):
\[ \cot\left(\frac{3\pi}{4}\right) = -1 \]
Substituting this into the equation gives:
\[ k = 1 + 2(-1) = 1 - 2 = -1 \]

Thus, the value of \( k \) is:
\[ \boxed{-1} \]