The value of `sec^2("tan"^(-1)3)+""cos"ec"^2(cot^(-1)2)` is equal to

Options

A. 5
B. 13
C. 15
D. 23

Answer

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

### Step 1: Define the angles
Let:
- \( \alpha = \tan^{-1}(3) \)
- \( \beta = \cot^{-1}(2) \)

### Step 2: Find \( \tan(\alpha) \) and \( \cot(\beta) \)
From the definitions:
- \( \tan(\alpha) = 3 \)
- \( \cot(\beta) = 2 \)

### Step 3: Find \( \sec^2(\alpha) \)
Using the identity \( \sec^2(\theta) = 1 + \tan^2(\theta) \):
\[ \sec^2(\alpha) = 1 + \tan^2(\alpha) = 1 + 3^2 = 1 + 9 = 10 \]

### Step 4: Find \( \sec^2(\beta) \)
Using the identity \( \sec^2(\theta) = 1 + \cot^2(\theta) \):
\[ \sec^2(\beta) = 1 + \cot^2(\beta) = 1 + 2^2 = 1 + 4 = 5 \]

### Step 5: Combine the results
Now, we can add the two results:
\[ \sec^2(\tan^{-1}(3)) + \sec^2(\cot^{-1}(2)) = 10 + 5 = 15 \]

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

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