If `2sin^(-1)((9-4x^(2))/(9+4x^(2)))+3cos^(-1)((12x)/(9+4x^(2)))+tan^(-1)((12x)/(9-4x^(2)))=lambda(pi)/(2) for x=(-17)/(13) "then" lambda`

Explanation

To solve the equation:

\[ 2\sin^{-1}\left(\frac{9-4x^2}{9+4x^2}\right) + 3\cos^{-1}\left(\frac{12x}{9+4x^2}\right) + \tan^{-1}\left(\frac{12x}{9-4x^2}\right) = \lambda \frac{\pi}{2} \]

for \( x = -\frac{17}{13} \), we will follow these steps:

### Step 1: Substitute the value of \( x \)

First, we substitute \( x = -\frac{17}{13} \) into the expressions inside the trigonometric functions.

\[ x = -\frac{17}{13} \]

Calculating \( 4x^2 \):

\[ 4x^2 = 4\left(-\frac{17}{13}\right)^2 = 4 \cdot \frac{289}{169} = \frac{1156}{169} \]

Now, calculate \( 9 + 4x^2 \) and \( 9 - 4x^2 \):

\[ 9 = \frac{1521}{169} \quad \text{(since } 9 = \frac{9 \cdot 169}{169}\text{)} \]

Thus,

\[ 9 + 4x^2 = \frac{1521 + 1156}{169} = \frac{2677}{169} \]

\[ 9 - 4x^2 = \frac{1521 - 1156}{169} = \frac{365}{169} \]

Now calculate \( 12x \):

\[ 12x = 12 \cdot \left(-\frac{17}{13}\right) = -\frac{204}{13} \]

### Step 2: Substitute into the inverse functions

Now we substitute these values into the inverse trigonometric functions:

1. **For \( \sin^{-1} \)**:

\[ \frac{9 - 4x^2}{9 + 4x^2} = \frac{1521 - 1156}{2677} = \frac{365}{2677} \]

2. **For \( \cos^{-1} \)**:

\[ \frac{12x}{9 + 4x^2} = \frac{-\frac{204}{13}}{\frac{2677}{169}} = -\frac{204 \cdot 169}{13 \cdot 2677} = -\frac{204 \cdot 13}{2677} = -\frac{2652}{2677} \]

3. **For \( \tan^{-1} \)**:

\[ \tan^{-1}\left(\frac{12x}{9 - 4x^2}\right) = \tan^{-1}\left(\frac{-\frac{204}{13}}{\frac{365}{169}}\right) = \tan^{-1}\left(-\frac{204 \cdot 169}{13 \cdot 365}\right) = \tan^{-1}\left(-\frac{204 \cdot 13}{365}\right) \]

### Step 3: Simplify the equation

Now we can rewrite the original equation using the above values.

Using the identity:

\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]

We can combine terms:

\[ 2\sin^{-1}\left(\frac{365}{2677}\right) + 3\cos^{-1}\left(-\frac{2652}{2677}\right) + \tan^{-1}\left(-\frac{204 \cdot 13}{365}\right) = \lambda \frac{\pi}{2} \]

### Step 4: Solve for \( \lambda \)

After substituting and simplifying, we find that:

\[ \lambda = 5 \]

### Final Answer:

Thus, the value of \( \lambda \) is:

\[ \lambda = 5 \]