What is the value of `sec^(2)(tan^(-1)2)`

Options

A. 1
B. 4
C. 5
D. 3

Answer

To find the value of \( \sec^2(\tan^{-1}(2)) \), we can follow these steps:

### Step 1: Define the angle
Let \( \theta = \tan^{-1}(2) \). This means that \( \tan(\theta) = 2 \).

### Step 2: Use the identity for secant
We know that the relationship between secant and tangent is given by the identity:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]

### Step 3: Substitute the value of tangent
Substituting \( \tan(\theta) = 2 \) into the identity:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) = 1 + (2)^2 \]

### Step 4: Calculate \( \tan^2(\theta) \)
Calculating \( (2)^2 \):
\[ (2)^2 = 4 \]

### Step 5: Complete the calculation
Now substitute back into the equation:
\[ \sec^2(\theta) = 1 + 4 = 5 \]

### Conclusion
Thus, the value of \( \sec^2(\tan^{-1}(2)) \) is:
\[ \sec^2(\tan^{-1}(2)) = 5 \]