Let `lambda=sec^2(tan^(-1)2)+cosec^2(cot^(-1)3)+2` then the value of `17lambda^2+7` is ……

Answer

To solve the problem, we need to find the value of \( \lambda = \sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3) + 2 \) and then compute \( 17\lambda^2 + 7 \).

### Step-by-step Solution:

1. **Calculate \( \sec^2(\tan^{-1} 2) \)**:
- Let \( a = \tan^{-1}(2) \).
- Then, \( \tan(a) = 2 \).
- Using the identity \( \sec^2(a) = 1 + \tan^2(a) \):
\[ \sec^2(a) = 1 + \tan^2(a) = 1 + 2^2 = 1 + 4 = 5. \]

2. **Calculate \( \csc^2(\cot^{-1} 3) \)**:
- Let \( b = \cot^{-1}(3) \).
- Then, \( \cot(b) = 3 \) implies \( \tan(b) = \frac{1}{3} \).
- Using the identity \( \csc^2(b) = 1 + \cot^2(b) \):
\[ \csc^2(b) = 1 + \cot^2(b) = 1 + 3^2 = 1 + 9 = 10. \]

3. **Combine the results to find \( \lambda \)**:
- Now substituting the values we found:
\[ \lambda = \sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3) + 2 = 5 + 10 + 2 = 17. \]

4. **Calculate \( 17\lambda^2 + 7 \)**:
- First, find \( \lambda^2 \):
\[ \lambda^2 = 17^2 = 289. \]
- Now substitute into the expression:
\[ 17\lambda^2 + 7 = 17 \times 289 + 7. \]
- Calculate \( 17 \times 289 \):
\[ 17 \times 289 = 4913. \]
- Finally, add 7:
\[ 4913 + 7 = 4920. \]

### Final Answer:
The value of \( 17\lambda^2 + 7 \) is \( \boxed{4920} \).