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

Options

A. 5
B. 10
C. 11
D. 15

Explanation

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

### Step 1: Find \( \tan^2(\sec^{-1} 2) \)

Let \( \theta = \sec^{-1}(2) \). By definition, this means that \( \sec(\theta) = 2 \).

Using the identity \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we have:
\[ \cos(\theta) = \frac{1}{2} \]

Now, we can find \( \sin(\theta) \) using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Substituting \( \cos(\theta) \):
\[ \sin^2(\theta) + \left(\frac{1}{2}\right)^2 = 1 \]
\[ \sin^2(\theta) + \frac{1}{4} = 1 \]
\[ \sin^2(\theta) = 1 - \frac{1}{4} = \frac{3}{4} \]

Now, we can find \( \tan(\theta) \):
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\sqrt{\frac{3}{4}}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \cdot 2 = \sqrt{3} \]

Thus, we have:
\[ \tan^2(\theta) = (\sqrt{3})^2 = 3 \]

### Step 2: Find \( \cot^2(\csc^{-1} 3) \)

Let \( \phi = \csc^{-1}(3) \). By definition, this means that \( \csc(\phi) = 3 \).

Using the identity \( \csc(\phi) = \frac{1}{\sin(\phi)} \), we have:
\[ \sin(\phi) = \frac{1}{3} \]

Now, we can find \( \cos(\phi) \) using the Pythagorean identity:
\[ \sin^2(\phi) + \cos^2(\phi) = 1 \]
Substituting \( \sin(\phi) \):
\[ \left(\frac{1}{3}\right)^2 + \cos^2(\phi) = 1 \]
\[ \frac{1}{9} + \cos^2(\phi) = 1 \]
\[ \cos^2(\phi) = 1 - \frac{1}{9} = \frac{8}{9} \]

Now, we can find \( \cot(\phi) \):
\[ \cot(\phi) = \frac{\cos(\phi)}{\sin(\phi)} = \frac{\sqrt{\frac{8}{9}}}{\frac{1}{3}} = \frac{\sqrt{8}}{3} \cdot 3 = \sqrt{8} = 2\sqrt{2} \]

Thus, we have:
\[ \cot^2(\phi) = (2\sqrt{2})^2 = 8 \]

### Step 3: Combine the results

Now we can combine the results from Step 1 and Step 2:
\[ \tan^2(\sec^{-1} 2) + \cot^2(\csc^{-1} 3) = 3 + 8 = 11 \]

### Final Answer
The value of \( \tan^2(\sec^{-1} 2) + \cot^2(\csc^{-1} 3) \) is \( \boxed{11} \).