If a^2·sec^2x − b^2·tan^2x = c^2 for real x (with a^2 ≠ b^2), then what is the value of sec^2x + tan^2x in terms of a, b and c?

Introduction / Context:
This problem checks your understanding of trigonometric identities and algebraic manipulation with parameters. You are given a relationship involving sec^2x and tan^2x with coefficients a^2 and b^2, and you are asked to find the sum sec^2x + tan^2x in terms of a, b and c. Questions of this type are common in aptitude exams to test how well you can combine identities with symbolic algebra.

Given Data / Assumptions:

• Equation: a^2·sec^2x − b^2·tan^2x = c^2.
• a, b and c are real parameters, with a^2 ≠ b^2.
• Standard identity: sec^2x − tan^2x = 1.
• x is real and sec x, tan x are defined.

Concept / Approach:
Introduce variables S = sec^2x and T = tan^2x. You then have one equation from the problem and another from the identity sec^2x − tan^2x = 1. Solving these two equations simultaneously gives S and T in terms of a, b and c. Finally, compute S + T and simplify the algebraic fraction.

Step-by-Step Solution:
Let S = sec^2x and T = tan^2x. Given: a^2·S − b^2·T = c^2. Identity: S − T = 1. From the identity, S = 1 + T. Substitute into a^2·S − b^2·T = c^2: a^2(1 + T) − b^2T = c^2. Expand: a^2 + a^2T − b^2T = c^2. Group T terms: (a^2 − b^2)T = c^2 − a^2. Thus T = (c^2 − a^2) / (a^2 − b^2). Now S = 1 + T = 1 + (c^2 − a^2)/(a^2 − b^2). Write 1 as (a^2 − b^2)/(a^2 − b^2): S = (a^2 − b^2 + c^2 − a^2)/(a^2 − b^2) = (c^2 − b^2)/(a^2 − b^2). Now compute S + T: S + T = (c^2 − b^2)/(a^2 − b^2) + (c^2 − a^2)/(a^2 − b^2). Combine numerators: S + T = (2c^2 − a^2 − b^2)/(a^2 − b^2). Multiply numerator and denominator by −1 to match option format: S + T = (a^2 + b^2 − 2c^2)/(b^2 − a^2).
Verification / Alternative check:
You can check by taking specific numerical values for a, b, c and solving the original system numerically for x to verify that sec^2x + tan^2x matches the formula. This is a good way to confirm that there are no sign mistakes in the algebra.

Why Other Options Are Wrong:
Options that use b^2 + a^2 in the denominator or incorrect sign patterns in the numerator do not satisfy the pair of equations when you substitute back. They usually come from mixing up the identity sec^2x − tan^2x = 1 or from sign errors when combining fractions.

Common Pitfalls:
A common error is to assume sec^2x + tan^2x also has a simple identity, which it does not. Another frequent mistake is forgetting to use S − T = 1, thus leaving the system underdetermined. Careful algebraic manipulation and correct use of the identity are essential.

Final Answer:
The correct value of sec^2x + tan^2x is (a^2 + b^2 − 2c^2) / (b^2 − a^2).