Prove the Identities?

( Sec8A-1)/( Sec4A-1)=Tan8A/Tan2A

(sec 8A - 1) / (Sec 4A - 1)  Taking LCM we get

  [(1 - cos 8A) * cos 4A] / [(1 - cos 4A) * cos 8A]

 =  (2sin²4A.cos 4A) / (2sin²2A.cos 8A)  [ (1 - cos 8A) = 2sin²4A and (1 - cos 4A) = 2sin²2A]

  = (2sin 4A.cos 4A.sin 4A) / (2sin 2A.sin 2A.cos8A)

 = (sin 8A.2sin 2A.cos 2A) / (2sin 2A.sin 2A.cos 8A)  [2sin4A.cos4A = sin8A and sin4A = 2sin2A.cos2A]

 = (sin 8A) / (cos 8A) * (cos 4A) / (sin 4A)

  = (tan 8A) / (tan 2A)

Hence Proved.