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Prove the trigonometric identity (tan A + sec A - 1)/(tan A - sec A + 1) = (1 + sin A)/cos A. Which conclusion is correct?

1. The identity is true for all admissible A.
2. The identity holds only for A = 45 deg.
3. The identity is false in general.
4. Both sides are undefined for every A.

Correct answer: The identity is true for all admissible A.

Solution

Using 1 = sec² A - tan² A = (sec A + tan A)(sec A - tan A), the numerator factors so the ratio simplifies to (sec A + tan A) = (1 + sin A)/cos A.

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