`sec^(2)(tan^(-1)2) +"cosec"^(2)(cot^(-1)3)=`

Options

A. 13
B. 14
C. 15
D. 16

Answer

To solve the expression \( \sec^2(\tan^{-1}2) + \csc^2(\cot^{-1}3) \), we will use the trigonometric identities for secant and cosecant functions.

### Step-by-Step Solution:

1. **Identify the angles**:
Let \( \theta_1 = \tan^{-1}(2) \) and \( \theta_2 = \cot^{-1}(3) \).

2. **Use the identity for secant**:
We know that:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]
Therefore, for \( \theta_1 \):
\[ \sec^2(\tan^{-1}(2)) = 1 + \tan^2(\tan^{-1}(2)) \]
Since \( \tan(\tan^{-1}(2)) = 2 \):
\[ \sec^2(\tan^{-1}(2)) = 1 + 2^2 = 1 + 4 = 5 \]

3. **Use the identity for cosecant**:
We also know that:
\[ \csc^2(\theta) = 1 + \cot^2(\theta) \]
Therefore, for \( \theta_2 \):
\[ \csc^2(\cot^{-1}(3)) = 1 + \cot^2(\cot^{-1}(3)) \]
Since \( \cot(\cot^{-1}(3)) = 3 \):
\[ \csc^2(\cot^{-1}(3)) = 1 + 3^2 = 1 + 9 = 10 \]

4. **Combine the results**:
Now, we can 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 \( 15 \).

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