What is the value of `(cos^2A)/(cos^2B)`, if `tan^2` A = 1 +2 `tan^2` B?

Options

A. `sqrt3`
B. `1/2`
C. `1/sqrt3`
D. 1

Answer

To find the value of \(\frac{\cos^2 A}{\cos^2 B}\) given that \(\tan^2 A = 1 + 2 \tan^2 B\), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \tan^2 A = 1 + 2 \tan^2 B \]

### Step 2: Add 1 to both sides
Adding 1 to both sides gives:
\[ 1 + \tan^2 A = 1 + (1 + 2 \tan^2 B) \]
This simplifies to:
\[ 1 + \tan^2 A = 2 + 2 \tan^2 B \]

### Step 3: Use the identity for secant
Using the identity \(1 + \tan^2 \theta = \sec^2 \theta\), we can rewrite the left side:
\[ \sec^2 A = 2 + 2 \tan^2 B \]

### Step 4: Factor the right side
We can factor out a 2 from the right side:
\[ \sec^2 A = 2(1 + \tan^2 B) \]

### Step 5: Apply the identity again
Using the identity again on the right side:
\[ \sec^2 A = 2 \sec^2 B \]

### Step 6: Relate secant to cosine
Recall that \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\). Therefore, we can write:
\[ \frac{1}{\cos^2 A} = 2 \cdot \frac{1}{\cos^2 B} \]

### Step 7: Cross-multiply
Cross-multiplying gives:
\[ \cos^2 B = 2 \cos^2 A \]

### Step 8: Rearranging the equation
This can be rearranged to find \(\frac{\cos^2 A}{\cos^2 B}\):
\[ \frac{\cos^2 A}{\cos^2 B} = \frac{1}{2} \]

### Conclusion
Thus, the value of \(\frac{\cos^2 A}{\cos^2 B}\) is:
\[ \frac{1}{2} \]