Prove the Identity : \[ (\sec x\sec y+\tan x\tan y)^2 – (\sec x\tan y+\tan x\sec y)^2 =1 \]

Solution:

\[ =(\sec x\sec y+\tan x\tan y+\sec x\tan y+\tan x\sec y) \]

\[ \times (\sec x\sec y+\tan x\tan y-\sec x\tan y-\tan x\sec y) \]

\[ = (\sec x+\tan x)(\sec y+\tan y) \]

\[ \times (\sec x-\tan x)(\sec y-\tan y) \]

\[ = (\sec^2 x-\tan^2 x) (\sec^2 y-\tan^2 y) \]

\[ =1 \times 1 \]

\[ =1 \]

Hence proved.