`sec^(2) (tan^(-1) 2) + cosec^(2) (cot^(-1) 3)` is equal to

Options

A. 5
B. 13
C. 15
D. 6

Explanation

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

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

### Step 2: Find \( \sec^2(\alpha) \)
From the definition of the tangent function, we have:
\[ \tan(\alpha) = 2 \implies \frac{\text{opposite}}{\text{adjacent}} = \frac{2}{1} \]
Using the Pythagorean theorem to find the hypotenuse:
\[ \text{hypotenuse} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]
Now, we can find \( \sec^2(\alpha) \):
\[ \sec^2(\alpha) = 1 + \tan^2(\alpha) = 1 + 2^2 = 1 + 4 = 5 \]

### Step 3: Find \( \csc^2(\beta) \)
From the definition of the cotangent function, we have:
\[ \cot(\beta) = 3 \implies \frac{\text{adjacent}}{\text{opposite}} = \frac{3}{1} \]
Using the Pythagorean theorem to find the hypotenuse:
\[ \text{hypotenuse} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]
Now, we can find \( \csc^2(\beta) \):
\[ \csc^2(\beta) = 1 + \cot^2(\beta) = 1 + 3^2 = 1 + 9 = 10 \]

### Step 4: Add the results
Now, we can add the two results:
\[ \sec^2(\tan^{-1}(2)) + \csc^2(\cot^{-1}(3)) = 5 + 10 = 15 \]

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

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