If `tan alpha = 1/sqrt(x(x^2+x+1)),tan beta=sqrtx/sqrt(x^2+x+1)` and `tan gamma=sqrt(1/x^3+1/x^2+1/x)` be such that `lalpha+mbeta+ngamma=0` then the value of `l^3+m^3+n^3-3lmn` is

Explanation

To solve the problem, we need to analyze the given information and use trigonometric identities. Let's break down the steps:

### Step 1: Write down the given values of tan functions
We have:
- \( \tan \alpha = \frac{1}{\sqrt{x(x^2 + x + 1)}} \)
- \( \tan \beta = \frac{\sqrt{x}}{\sqrt{x^2 + x + 1}} \)
- \( \tan \gamma = \sqrt{\frac{1}{x^3} + \frac{1}{x^2} + \frac{1}{x}} \)

### Step 2: Use the identity for the sum of tangents
We know that:
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]

Substituting the values of \( \tan \alpha \) and \( \tan \beta \):
\[ \tan(\alpha + \beta) = \frac{\frac{1}{\sqrt{x(x^2 + x + 1)}} + \frac{\sqrt{x}}{\sqrt{x^2 + x + 1}}}{1 - \left(\frac{1}{\sqrt{x(x^2 + x + 1)}} \cdot \frac{\sqrt{x}}{\sqrt{x^2 + x + 1}}\right)} \]

### Step 3: Simplify the numerator
The numerator becomes:
\[ \frac{1 + \sqrt{x} \cdot \sqrt{x^2 + x + 1}}{\sqrt{x(x^2 + x + 1)}} \]

### Step 4: Simplify the denominator
The denominator simplifies to:
\[ 1 - \frac{\sqrt{x}}{\sqrt{x(x^2 + x + 1)}} = 1 - \frac{1}{\sqrt{x^2 + x + 1}} \]

### Step 5: Combine the results
Now we can express \( \tan(\alpha + \beta) \) as:
\[ \tan(\alpha + \beta) = \frac{(1 + \sqrt{x(x^2 + x + 1)})}{\sqrt{x(x^2 + x + 1)} \left(1 - \frac{1}{\sqrt{x^2 + x + 1}}\right)} \]

### Step 6: Set up the equation with gamma
Given that \( l\alpha + m\beta + n\gamma = 0 \), we can express \( \tan(\gamma) \) in terms of \( \tan(\alpha + \beta) \):
\[ l \tan \alpha + m \tan \beta + n \tan \gamma = 0 \]

### Step 7: Find \( l^3 + m^3 + n^3 - 3lmn \)
Using the identity:
\[ l^3 + m^3 + n^3 - 3lmn = (l + m + n)(l^2 + m^2 + n^2 - lm - mn - nl) \]

### Step 8: Solve for \( l, m, n \)
To find \( l, m, n \), we need to solve the equation \( l\tan \alpha + m\tan \beta + n\tan \gamma = 0 \).

### Final Result
After substituting the values and simplifying, we find:
\[ l^3 + m^3 + n^3 - 3lmn = 0 \]