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)] [secx sec2 y + tanx tan2 y + 2(sec x sec y) (tan x tan y)] – [secx tan2 y + tanx sec2 y + 2(secx tan2 y) (tan x sec y)]

secx sec2 y – secx tan2 y + tanx tan2 y – tanx sec2 y

secx(sec2 y – tan2 y) + tanx(tan2 y – sec2 y)

secx(sec2 y – tan2 y) – tanx(sec2 y – tan2 y)

As we know, secx – tanx = 1.

secx × 1 – tanx × 1

secx – tanx

1 = RHS

∴ LHS = RHS

Thus proved.



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