The value of
`sec^(2)(tan^(-1)2)+cosec^(2)(cot^(-1)3)` is

Options

A. 5
B. 10
C. 15
D. 20

Explanation

To solve the expression \( \sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3) \), we can follow these steps:

### Step 1: Understanding the Functions
We start by recalling the definitions of the secant and cosecant functions in terms of tangent and cotangent:
- \( \sec^2(\theta) = 1 + \tan^2(\theta) \)
- \( \csc^2(\phi) = 1 + \cot^2(\phi) \)

### Step 2: Applying the Definitions
Let \( \theta = \tan^{-1}(2) \) and \( \phi = \cot^{-1}(3) \). Then we can rewrite the expression:
\[ \sec^2(\tan^{-1} 2) = 1 + \tan^2(\tan^{-1} 2) \]
\[ \csc^2(\cot^{-1} 3) = 1 + \cot^2(\cot^{-1} 3) \]

### Step 3: Evaluating the Tangent and Cotangent
From the definitions:
- \( \tan(\tan^{-1}(2)) = 2 \), so \( \tan^2(\tan^{-1}(2)) = 2^2 = 4 \).
- \( \cot(\cot^{-1}(3)) = 3 \), so \( \cot^2(\cot^{-1}(3)) = 3^2 = 9 \).

### Step 4: Substituting Back into the Expression
Now substituting these values back into our expressions:
\[ \sec^2(\tan^{-1} 2) = 1 + 4 = 5 \]
\[ \csc^2(\cot^{-1} 3) = 1 + 9 = 10 \]

### Step 5: Adding the Two Results
Now we add the two results together:
\[ \sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3) = 5 + 10 = 15 \]

### Final Answer
Thus, the value of \( \sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3) \) is \( \boxed{15} \).

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