Prove the given identities. 1. cos^4(x)-sin^4(x) = cos(2x) 2. (1-cos(X))/(1+cos(X))=tan^2 (x/2)

Kade Reese 2022-07-26 Answered
Prove the given identities.
1. cos 4 ( x ) sin 4 ( x ) = cos ( 2 x )
2. 1 cos ( X ) 1 + cos ( X ) = tan 2 ( x 2 )
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Answers (1)

Cheyanne Charles
Answered 2022-07-27 Author has 13 answers
1) cos 4 x sin 4 x = ( cos 2 x sin 2 x ) ( cos 2 x + sin 2 x )
we have ( cos 2 x + sin 2 x ) = 1   a n d   cos 2 x = 2 cos 2 x 1
= cos 2 x sin 2 x
= cos 2 x ( 1 cos 2 x )
= 2 cos 2 x 1
= cos ( 2 x )
2) We have the trignometric relations
1 + cos ( 2 t ) = 2 cos 2 t
1 cos ( 2 t ) = 2 sin 2 t
Putting 2t = x, we get
1 + cos ( x ) = 2 cos 2 ( x / 2 ) 1
1 cos ( x ) = 2 sin 2 ( x / 2 ) 2
2 / 1 tan 2 ( x / 2 )
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