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]

= (2sin^{2}4A.cos 4A) / (2sin^{2}2A.cos 8A) [ (1 - cos 8A) = 2sin^{2}4A and (1 - cos 4A) = 2sin^{2}2A]

= (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.

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