The sum of all the solutions of `cottheta=sin2theta(theta!=npi, n in t ege r)` , `0lt=thetalt=pi,` is `(3pi)/2` (b) `pi` (c) 3`pi/4` (d) `2pi`

Options

A. `3pi//2`
B. `pi`
C. `3pi//4`
D. `2pi`

Explanation

To solve the equation \( \cot \theta = \sin 2\theta \) for \( 0 < \theta < \pi \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \cot \theta = \sin 2\theta \]
We know that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \sin 2\theta = 2 \sin \theta \cos \theta \). Therefore, we can rewrite the equation as:
\[ \frac{\cos \theta}{\sin \theta} = 2 \sin \theta \cos \theta \]

### Step 2: Clear the fraction
To eliminate the fraction, we multiply both sides by \( \sin \theta \) (noting that \( \sin \theta \neq 0 \) in the interval \( (0, \pi) \)):
\[ \cos \theta = 2 \sin^2 \theta \cos \theta \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation:
\[ \cos \theta - 2 \sin^2 \theta \cos \theta = 0 \]
Factoring out \( \cos \theta \):
\[ \cos \theta (1 - 2 \sin^2 \theta) = 0 \]

### Step 4: Set each factor to zero
We have two cases to consider:
1. \( \cos \theta = 0 \)
2. \( 1 - 2 \sin^2 \theta = 0 \)

### Step 5: Solve \( \cos \theta = 0 \)
The equation \( \cos \theta = 0 \) gives us:
\[ \theta = \frac{\pi}{2} \]
This solution is valid since \( \frac{\pi}{2} \) is in the interval \( (0, \pi) \).

### Step 6: Solve \( 1 - 2 \sin^2 \theta = 0 \)
Rearranging gives:
\[ 2 \sin^2 \theta = 1 \quad \Rightarrow \quad \sin^2 \theta = \frac{1}{2} \quad \Rightarrow \quad \sin \theta = \pm \frac{1}{\sqrt{2}} \]
The solutions for \( \sin \theta = \frac{1}{\sqrt{2}} \) are:
\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{3\pi}{4} \]
Both solutions are valid since they lie in the interval \( (0, \pi) \).

### Step 7: List all solutions
The solutions we found are:
1. \( \theta = \frac{\pi}{2} \)
2. \( \theta = \frac{\pi}{4} \)
3. \( \theta = \frac{3\pi}{4} \)

### Step 8: Calculate the sum of the solutions
Now, we sum all the solutions:
\[ \text{Sum} = \frac{\pi}{2} + \frac{\pi}{4} + \frac{3\pi}{4} \]
Finding a common denominator (which is 4):
\[ \text{Sum} = \frac{2\pi}{4} + \frac{\pi}{4} + \frac{3\pi}{4} = \frac{2\pi + \pi + 3\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2} \]

### Final Answer
The sum of all the solutions is:
\[ \frac{3\pi}{2} \]
Thus, the correct option is (a) \( \frac{3\pi}{2} \).

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