If `cos 2A = (7)/(19)`, what is the value of `(1)/(cos^(2)A-sin^(2)A)` ?

Options

A. `0.18`
B. `0.37`
C. `0.74`
D. `2.71`

Explanation

To solve the problem, we need to find the value of \( \frac{1}{\cos^2 A - \sin^2 A} \) given that \( \cos 2A = \frac{7}{19} \).

### Step-by-Step Solution:

1. **Use the double angle formula for cosine**:
The double angle formula states that:
\[ \cos 2A = \cos^2 A - \sin^2 A \]
We can substitute the given value into this formula.

2. **Substitute the given value**:
From the problem, we know:
\[ \cos 2A = \frac{7}{19} \]
Therefore, we can write:
\[ \cos^2 A - \sin^2 A = \frac{7}{19} \]

3. **Find the reciprocal**:
We need to find \( \frac{1}{\cos^2 A - \sin^2 A} \). Since we have already established that \( \cos^2 A - \sin^2 A = \frac{7}{19} \), we take the reciprocal:
\[ \frac{1}{\cos^2 A - \sin^2 A} = \frac{1}{\frac{7}{19}} = \frac{19}{7} \]

4. **Convert to decimal form**:
Now, we convert \( \frac{19}{7} \) into decimal form:
\[ \frac{19}{7} \approx 2.714285714285714 \]
Rounding this to two decimal places gives us:
\[ \approx 2.71 \]

### Final Answer:
Thus, the value of \( \frac{1}{\cos^2 A - \sin^2 A} \) is approximately \( 2.71 \).