Saved Bookmarks
| 1. |
Prove the identity: (sec x sec y + tan x tan y)2 – (sec x tan y + tan x sec y)2 = 1 |
|
Answer» Let us consider the LHS: (sec x sec y + tan x tan y)2 – (sec x tan y + tan x sec y)2 By expanding the above equation we get, [(sec x sec y)2 + (tan x tan y)2 + 2(sec x sec y) (tan x tan y)] – [(sec x tan y)2 + (tan x sec y)2 + 2(sec x tan y) (tan x sec y)] [sec2 x sec2 y + tan2 x tan2 y + 2(sec x sec y) (tan x tan y)] – [sec2 x tan2 y + tan2 x sec2 y + 2(sec2 x tan2 y) (tan x sec y)] sec2 x sec2 y – sec2 x tan2 y + tan2 x tan2 y – tan2 x sec2 y sec2 x(sec2 y – tan2 y) + tan2 x(tan2 y – sec2 y) sec2 x(sec2 y – tan2 y) – tan2 x(sec2 y – tan2 y) As we know, sec2 x – tan2 x = 1. sec2 x × 1 – tan2 x × 1 sec2 x – tan2 x 1 = RHS ∴ LHS = RHS Thus proved. |
|