If ` tan^(-1) 4 + tan ^(-1) 5 = cot^(-1) lambda " , then find " 'lambda'`

Answer

To solve the equation \( \tan^{-1}(4) + \tan^{-1}(5) = \cot^{-1}(\lambda) \), we will use some trigonometric identities.

### Step 1: Use the identity for the sum of inverse tangents
We know that:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]
for \( ab < 1 \) or we can adjust it for \( ab > 1 \) by adding \( \pi \) if necessary.

In our case, \( a = 4 \) and \( b = 5 \).

### Step 2: Calculate \( a + b \) and \( ab \)
Calculate:
\[ a + b = 4 + 5 = 9 \]
\[ ab = 4 \cdot 5 = 20 \]

### Step 3: Apply the identity
Since \( ab = 20 > 1 \), we need to use the identity:
\[ \tan^{-1}(4) + \tan^{-1}(5) = \tan^{-1}\left(\frac{9}{1 - 20}\right) + \pi \]
Thus,
\[ \tan^{-1}(4) + \tan^{-1}(5) = \tan^{-1}\left(\frac{9}{-19}\right) + \pi \]

### Step 4: Relate to cotangent
Since \( \cot^{-1}(\lambda) = \frac{\pi}{2} - \tan^{-1}(\lambda) \), we can express:
\[ \cot^{-1}(\lambda) = \frac{\pi}{2} - \tan^{-1}\left(\frac{9}{-19}\right) \]
This implies:
\[ \lambda = -\frac{19}{9} \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = -\frac{19}{9} \]

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