Prove the identity: (1 - sec(x))/(1 + sec(x)) = (cos(x) - 1)/(cos(x) + 1)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to prove that: (1 - sec(x))/(1 + sec(x)) = (cos(x) - 1)/(cos(x) + 1)

Let's start from the left hand side

(1 - sec(x))/(1 + sec(x))

use the fact that sec x = 1/cos x

=> (1 - (1/cos x))/(1 + (1/cos x))

=> [(cos x - 1)/cos x]/[(cos x + 1)/cos x]

=> (cos x - 1)/(cos x + 1)

which is the right hand side

This proves that (1 - sec(x))/(1 + sec(x)) = (cos(x) - 1)/(cos(x) + 1)

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