Statement 1: `tan^(2)(sec^(-1)2)+cot^(2)(cosec^(-1)3)=11`.
Statement -2 `:tan^(2)theta+sec^(2)theta=1=cot^(2)theta+cosec^(2)theta`

Options

A. Statement -1 is true, Statement -2 is true and Statement -2 is correct explanation for Statement -1
B. Statement -1 is true, Statement -2 is true and Statement -2 is not correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false Statement -2 is true

Explanation

To solve the given question, we need to analyze both statements step by step.

### Statement 1:
We need to prove that:
\[ \tan^2(\sec^{-1}(2)) + \cot^2(\csc^{-1}(3)) = 11 \]

**Step 1: Calculate \(\tan^2(\sec^{-1}(2))\)**
Let \( x = \sec^{-1}(2) \). By definition, this means:
\[ \sec(x) = 2 \]
Using the identity \(\sec^2(x) = 1 + \tan^2(x)\), we can derive:
\[ \tan^2(x) = \sec^2(x) - 1 = 2^2 - 1 = 4 - 1 = 3 \]
Thus,
\[ \tan^2(\sec^{-1}(2)) = 3 \]

**Step 2: Calculate \(\cot^2(\csc^{-1}(3))\)**
Let \( y = \csc^{-1}(3) \). By definition, this means:
\[ \csc(y) = 3 \]
Using the identity \(\csc^2(y) = 1 + \cot^2(y)\), we can derive:
\[ \cot^2(y) = \csc^2(y) - 1 = 3^2 - 1 = 9 - 1 = 8 \]
Thus,
\[ \cot^2(\csc^{-1}(3)) = 8 \]

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

### Conclusion for Statement 1:
Since we have shown that:
\[ \tan^2(\sec^{-1}(2)) + \cot^2(\csc^{-1}(3)) = 11 \]
Statement 1 is **True**.

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### Statement 2:
We need to analyze whether:
\[ \tan^2(\theta) + \sec^2(\theta) = 1 = \cot^2(\theta) + \csc^2(\theta) \]

**Step 1: Analyze \(\tan^2(\theta) + \sec^2(\theta)\)**
Using the identity:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]
This means:
\[ \tan^2(\theta) + \sec^2(\theta) = \tan^2(\theta) + (1 + \tan^2(\theta)) = 2\tan^2(\theta) + 1 \]
This cannot equal 1 unless \(\tan^2(\theta) = 0\), which is not generally true for all \(\theta\).

**Step 2: Analyze \(\cot^2(\theta) + \csc^2(\theta)\)**
Using the identity:
\[ \csc^2(\theta) = 1 + \cot^2(\theta) \]
This means:
\[ \cot^2(\theta) + \csc^2(\theta) = \cot^2(\theta) + (1 + \cot^2(\theta)) = 2\cot^2(\theta) + 1 \]
This also cannot equal 1 unless \(\cot^2(\theta) = 0\), which is again not generally true for all \(\theta\).

### Conclusion for Statement 2:
Since both parts of Statement 2 are not true, we conclude that Statement 2 is **False**.

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### Final Conclusion:
- Statement 1 is **True**.
- Statement 2 is **False**.