If `a^(2)sec^(2)x-b^(2)tan^(2)x=c^(2)` then the value of `(sec^(2)x+tan^(2)x)` is equal to (assume `b^(2)nea^(2))`

Options

A. `(a^(2)-a^(2)+2c^(2))/(b^(2)+a^(2))`
B. `(b^(2)+a^(2)-2c^(2))/(b^(2)-a^(2))`
C. `(b^(2)-a^(2)-2c^(2))/(b^(2)+a^(2))`
D. `(b^(2)-a^(2))/(b^(2)+a^(2)+2c^(2))`

Answer

To solve the equation \( a^2 \sec^2 x - b^2 \tan^2 x = c^2 \) and find the value of \( \sec^2 x + \tan^2 x \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ a^2 \sec^2 x - b^2 \tan^2 x = c^2 \]
Using the identity \( \sec^2 x = 1 + \tan^2 x \), we can substitute \( \sec^2 x \):
\[ a^2 (1 + \tan^2 x) - b^2 \tan^2 x = c^2 \]

### Step 2: Expand the Equation
Expanding the left side gives:
\[ a^2 + a^2 \tan^2 x - b^2 \tan^2 x = c^2 \]
This simplifies to:
\[ a^2 + (a^2 - b^2) \tan^2 x = c^2 \]

### Step 3: Isolate \( \tan^2 x \)
Rearranging the equation to isolate \( \tan^2 x \):
\[ (a^2 - b^2) \tan^2 x = c^2 - a^2 \]
Thus, we can express \( \tan^2 x \) as:
\[ \tan^2 x = \frac{c^2 - a^2}{a^2 - b^2} \]

### Step 4: Find \( \sec^2 x + \tan^2 x \)
Now, we need to find \( \sec^2 x + \tan^2 x \):
\[ \sec^2 x + \tan^2 x = (1 + \tan^2 x) + \tan^2 x = 1 + 2\tan^2 x \]
Substituting \( \tan^2 x \) from the previous step:
\[ \sec^2 x + \tan^2 x = 1 + 2 \left( \frac{c^2 - a^2}{a^2 - b^2} \right) \]

### Step 5: Simplify the Expression
Simplifying this expression:
\[ \sec^2 x + \tan^2 x = 1 + \frac{2(c^2 - a^2)}{a^2 - b^2} \]
Combining the terms gives:
\[ \sec^2 x + \tan^2 x = \frac{(a^2 - b^2) + 2(c^2 - a^2)}{a^2 - b^2} \]
This simplifies to:
\[ \sec^2 x + \tan^2 x = \frac{b^2 + a^2 - 2c^2}{a^2 - b^2} \]

### Final Answer
Thus, the value of \( \sec^2 x + \tan^2 x \) is:
\[ \sec^2 x + \tan^2 x = \frac{b^2 + a^2 - 2c^2}{a^2 - b^2} \]

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