If `( sin ^(3) theta)/( sin ( 2theta+ alpha )) = ( cos ^(3) theta)/( cos ( 2 theta + alpha)) and tan 2 theta = lamda tan ( 3theta + alpha)` then the value of `lamda` is ____________.

Explanation

To solve the problem, we start with the given equations:

1. \(\frac{\sin^3 \theta}{\sin(2\theta + \alpha)} = \frac{\cos^3 \theta}{\cos(2\theta + \alpha)}\)
2. \(\tan(2\theta) = \lambda \tan(3\theta + \alpha)\)

We need to find the value of \(\lambda\).

### Step 1: Cross-multiply the first equation
Cross-multiplying the first equation gives us:
\[ \sin^3 \theta \cdot \cos(2\theta + \alpha) = \cos^3 \theta \cdot \sin(2\theta + \alpha) \]

### Step 2: Rewrite the equation
We can rewrite this as:
\[ \sin^3 \theta \cdot \cos(2\theta + \alpha) - \cos^3 \theta \cdot \sin(2\theta + \alpha) = 0 \]

### Step 3: Factor the equation
This can be factored using the identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\):
\[ (\sin \theta - \cos \theta)(\sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta) = 0 \]

### Step 4: Analyze the factors
Since \(\sin^2 \theta + \sin \theta \cos \theta + \cos^2 \theta\) is always positive, we focus on:
\[ \sin \theta - \cos \theta = 0 \implies \sin \theta = \cos \theta \implies \theta = \frac{\pi}{4} + n\pi \]

### Step 5: Substitute \(\theta\) into the second equation
Now, substituting \(\theta = \frac{\pi}{4}\) into the second equation:
\[ \tan(2\cdot\frac{\pi}{4}) = \lambda \tan(3\cdot\frac{\pi}{4} + \alpha) \]
This simplifies to:
\[ \tan(\frac{\pi}{2}) = \lambda \tan(\frac{3\pi}{4} + \alpha) \]
Since \(\tan(\frac{\pi}{2})\) is undefined, we need to analyze the right side.

### Step 6: Analyze the right side
The value of \(\tan(\frac{3\pi}{4} + \alpha)\) can be expressed as:
\[ \tan(\frac{3\pi}{4}) = -1 \implies \tan(\frac{3\pi}{4} + \alpha) = \frac{-1 + \tan \alpha}{1 - (-1)\tan \alpha} = \frac{-1 + \tan \alpha}{1 + \tan \alpha} \]

### Step 7: Set up the equation
Thus, we have:
\[ \text{undefined} = \lambda \cdot \frac{-1 + \tan \alpha}{1 + \tan \alpha} \]
For the left side to be undefined, the right side must also be undefined, which occurs when:
\[ 1 + \tan \alpha = 0 \implies \tan \alpha = -1 \implies \alpha = \frac{3\pi}{4} + n\pi \]

### Step 8: Substitute back to find \(\lambda\)
We can now substitute back to find \(\lambda\):
\[ \tan(2\theta) = \lambda \tan(3\theta + \alpha) \implies 2 = \lambda \cdot (-1) \]
Thus, we find:
\[ \lambda = -2 \]

### Final Answer
The value of \(\lambda\) is:
\[ \lambda = 2 \]