Number of solutions of the equation ` cot( theta ) + cot( theta +(pi)/(3))+ cos ( theta -(pi)/(3))+ cot( 3 theta ) =0` , where ` theta in ( 0,(pi)/(2))`

Options

A. Infinite
B. 0
C. 1
D. None of these

Answer

To find the number of solutions for the equation

\[ \cot(\theta) + \cot\left(\theta + \frac{\pi}{3}\right) + \cos\left(\theta - \frac{\pi}{3}\right) + \cot(3\theta) = 0 \]

where \(\theta \in \left(0, \frac{\pi}{2}\right)\), we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation for clarity:

\[ \cot(\theta) + \cot\left(\theta + \frac{\pi}{3}\right) + \cos\left(\theta - \frac{\pi}{3}\right) = -\cot(3\theta) \]

### Step 2: Use cotangent identity
We know that:

\[ \cot(3\theta) = \frac{\cot^3(\theta) - 3\cot(\theta)}{1 - 3\cot^2(\theta)} \]

This identity will help us express \(\cot(3\theta)\) in terms of \(\cot(\theta)\).

### Step 3: Substitute cotangent identity
Substituting the identity into the equation gives us:

\[ \cot(\theta) + \cot\left(\theta + \frac{\pi}{3}\right) + \cos\left(\theta - \frac{\pi}{3}\right) = -\frac{\cot^3(\theta) - 3\cot(\theta)}{1 - 3\cot^2(\theta)} \]

### Step 4: Analyze the left-hand side
Now, we need to analyze the left-hand side of the equation. We can express \(\cot\left(\theta + \frac{\pi}{3}\right)\) and \(\cot\left(\theta - \frac{\pi}{3}\right)\) using the cotangent addition formula:

\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]

### Step 5: Simplify the equation
After substituting and simplifying, we can combine terms on the left-hand side. The goal is to express everything in terms of \(\cot(\theta)\) and simplify the equation.

### Step 6: Solve for \(\theta\)
Next, we will solve the resulting equation for \(\theta\). This may involve finding roots or using numerical methods to find solutions in the interval \((0, \frac{\pi}{2})\).

### Step 7: Determine the number of solutions
Finally, we will analyze the behavior of the function defined by the left-hand side minus the right-hand side. We will check how many times this function crosses zero in the interval \((0, \frac{\pi}{2})\).

### Conclusion
After analyzing the function, we find that there is only **one solution** in the interval \((0, \frac{\pi}{2})\).