What is (\sec^2\alpha+\tan \alpha \cdot \tan \beta-\tan^2\alpha)^2+(\tan \alpha-\tan \beta)^2-\sec^2\alpha \cdot \sec^2\beta equal to?

Explanation

Using \sec^2\alpha - \tan^2\alpha = 1, the first term becomes (1+\tan\alpha\tan\beta)^2. Expanding the first two terms: (1+2\tan\alpha\tan\beta+\tan^2\alpha\tan^2\beta) + (\tan^2\alpha-2\tan\alpha\tan\beta+\tan^2\beta) = 1+\tan^2\alpha+\tan^2\beta+\tan^2\alpha\tan^2\beta. This factors into (1+\tan^2\alpha)(1+\tan^2\beta) = \sec^2\alpha\sec^2\beta. Subtracting \sec^2\alpha\sec^2\beta leaves 0.

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