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Prove that cos 4x = 1 - 8 sin^2 x cos^2 x
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cos 4 x = 1 - 8 sin² x cos² x [ Proved ]
Step-by-step explanation:
Given :
cos 4 x = 1 - 8 sin² x cos² x
L.H.S. = cos 4 x
= > cos 2 ( 2 x )
Using multiple angle formula :
i.e. cos 2 x = 2 cos² x - 1
= > cos 2 ( 2 x )
= > cos 2 ( 2 x )
= > 2 ( 2 cos² 2 x - 1 ) - 1
= > 2 ( 2 cos² x - 1 )² - 1
= > 2 ( 4 cos⁴ x - 4 cos² x + 1 ) - 1
= > 8 cos⁴ x - 8 cos² + 2 - 1
= > 1 - 8 cos² x ( 1 - cos² x )
We know :
1 - cos² x = sin² x
= > 1 - 8 cos² x . sin² x
Since L.H.S. = R.H.S
Hence proved.
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