Question

. Prove that
cos^2 2x - cos^2 6x = sin 4x sin 8x​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

cos² x - cos² 6 x = sin 4 x. sin 8 x  [ Proved ]

Step-by-step explanation:

Given :

cos² x - cos² 6 x = sin 4 x. sin 8 x

L.H.S. = cos² x - cos² 6 x

Using identity a² - b² = ( a + b ) ( a - b )

= > ( cos 2 x + cos 6 x ) ( cos 2 x - cos 6 x )

Using sum and difference to product formula :

cos C + cos D = 2 cos ( ( C + D ) / 2 ) . cos ( ( ( C - D ) / 2 )

cos C - cos D =  - 2 sin ( ( C + D ) / 2 ) . sin ( ( ( C - D ) / 2 )

= > [ 2 cos ( 2 x + 6 x ) / 2. cos ( 2 x - 6 x ) / 2 ] [ - sin ( 2 x + 6 x ) / 2 . sin ( 2 x - 6 x ) / 2 ]

= > 2 cos 4 x . cos ( -2 x ) ( - 2 sin 4 x sin ( - 2 x ) )

We know :

cos ( - Ф ) = cos Ф and sin ( Ф ) = - sin Ф

= > 2 cos 4 x . cos ( 2 x ) ( 2 sin 4 x sin ( 2 x ) )

Using multiple angle formula :

i.e. 2 sin x cos x = sin 2 x

= > ( 2 cos 4 x .cos 4 x ) ( 2 sin 2 x . cos 2 x )

= > sin 8 x . sin 4 x

= R.H.S.

Since L.H.S. = R.H.S.

Hence proved.