What is the value of `sec^(2)tan^(-1)((5)/(11))`?

Options

A. 121/96
B. 211/921
C. 146/121
D. 267/121

Explanation

To find the value of \( \sec^2(\tan^{-1}(\frac{5}{11})) \), we can follow these steps:

### Step 1: Let \( \theta = \tan^{-1}(\frac{5}{11}) \)

This means that \( \tan(\theta) = \frac{5}{11} \).

### Step 2: Use the identity for secant squared

We know from trigonometric identities that:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]

### Step 3: Calculate \( \tan^2(\theta) \)

Since \( \tan(\theta) = \frac{5}{11} \), we can find \( \tan^2(\theta) \):
\[ \tan^2(\theta) = \left(\frac{5}{11}\right)^2 = \frac{25}{121} \]

### Step 4: Substitute \( \tan^2(\theta) \) into the secant squared identity

Now substitute \( \tan^2(\theta) \) into the secant squared identity:
\[ \sec^2(\theta) = 1 + \frac{25}{121} \]

### Step 5: Convert 1 into a fraction

To add the fractions, convert 1 into a fraction with a denominator of 121:
\[ 1 = \frac{121}{121} \]

### Step 6: Add the fractions

Now add the two fractions:
\[ \sec^2(\theta) = \frac{121}{121} + \frac{25}{121} = \frac{121 + 25}{121} = \frac{146}{121} \]

### Conclusion

Thus, the value of \( \sec^2(\tan^{-1}(\frac{5}{11})) \) is:
\[ \sec^2(\tan^{-1}(\frac{5}{11})) = \frac{146}{121} \]

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