Prove the Identity : \[ (1+\tan\alpha\tan\beta)^2 + (\tan\alpha-\tan\beta)^2 = \sec^2\alpha\sec^2\beta \]

Solution:

\[ (1+\tan\alpha\tan\beta)^2 + (\tan\alpha-\tan\beta)^2 \]

\[ = 1+\tan^2\alpha\tan^2\beta +2\tan\alpha\tan\beta \]

\[ + \tan^2\alpha+\tan^2\beta -2\tan\alpha\tan\beta \]

\[ = 1+\tan^2\alpha+\tan^2\beta +\tan^2\alpha\tan^2\beta \]

\[ = (1+\tan^2\alpha)(1+\tan^2\beta) \]

\[ = \sec^2\alpha\sec^2\beta \]

Hence proved.