What is (1+\tan \alpha \tan \beta)^2+(\tan \alpha-\tan \beta)^2 equal to?
- A. \tan^2\alpha \tan^2\beta
- B. \sec^2\alpha \sec^2\beta ✓
- C. \tan^2\alpha \cot^2\beta
- D. \sec^2\alpha \tan^2\beta
Correct Answer: B. \sec^2\alpha \sec^2\beta
Explanation
Expanding both terms: (1 + 2\tan \alpha \tan \beta + \tan^2 \alpha \tan^2 \beta) + (\tan^2 \alpha - 2\tan \alpha \tan \beta + \tan^2 \beta). The middle terms cancel, leaving 1 + \tan^2 \alpha + \tan^2 \beta + \tan^2 \alpha \tan^2 \beta. Factoring by grouping yields (1+\tan^2 \alpha)(1+\tan^2 \beta) = \sec^2 \alpha \sec^2 \beta.
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