Can you prove this identity of cos(4x)=1-8sin2x+8sin2x ?
Can you prove this identity of cos(4x)=1-8sin2x+8sin2x ?
Esteban Garcia , 8 Years ago
Grade 12th pass
Trigonometry> Can you prove this identity of cos(4x)=1-...
1 AnswersNandana
Last Activity: 8 Years ago
hi,
I think it’s not “1-8 sin2 x + 8 sin2 x “ = 1 , it is
‘ 1- 8 sin2 x cos2 x ‘ , the proof is this way –
cos (4x) = cos (2x+2x) = cos2 (2x) – sin2 (2x)
= [cos2 (x) – sin2 (x) ]2 – (2 sin(x) cos (x)) 2
= cos4 x + sin4 x – 2 cos2 (x) sin2 (x) – 4 sin2 x cos2 x
= cos4 x + sin4 x -6 cos2 (x) sin2 (x)
= cos4 x + sin4 x +2 cos2 (x) sin2 (x) -2 cos2 (x) sin2 (x)-6 cos2 (x) sin2 (x)
=(cos2 x + sin2 x)2 – 8 cos2 (x) sin2 (x)
= 1- 8 cos2 (x) sin2 (x)
thank you !
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