Can the following identity be verified: sec^4 x - sec^2 x = tan^4 x + tan^2 x?

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We are asked to verify the following identity:

sec^4 x - sec^2 x = tan^4 x + tan^2 x

=> sec^4 x - sec^2 x=   tan^2 x ( tan^2 x + 1)

=> sec^4 x - sec^2 x =  tan^2 x ( sec^2 x)

=> sec^4 x - sec^2 x =  sec^2 x -1 ( sec^2 x)

=> sec^4 x - sec^2 x =  sec^4 x - sec^2 x

The identity sec^4 x -sec^2 x = tan^4 x + tan^2 x can be verified.

 

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We need to verify that (sec x)^4 - (sec x)^2 = (tan x)^4 + (tan x)^2.

We know that (sin x)^2 + (cos x)^2 = 1

=> (sin x)^2 / (cos x)^2 + (cos x)^2/ (cos x)^2 = 1/(cos x)^2

=> (tan x)^2 + 1 = (sec x)^2

Starting with the left hand side:

(sec x)^4 - (sec x)^2

=> (sec x)^2[(sec x)^2 - 1]

=> [(tan x)^2 + 1][(tan x)^2 + 1 - 1]

=> [(tan x)^2 + 1][(tan x)^2]

=> (tan x)^4 + (tan x)^2

which is the right hand side.

This proves that (sec x)^4 - (sec x)^2 = (tan x)^4 + (tan x)^2.

Approved by eNotes Editorial Team
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