13. cos-2-cos2 6x = sin 4x sin 8x
L.H.S.= cos 2 2 x − cos 2 6 x
Using the trigonometric identities
cos A + cos B = 2 cos ( A + B 2 ) ⋅ cos ( A − B 2 ) cos A − cos B = − 2 sin ( A + B 2 ) ⋅ sin ( A − B 2 )
Simplifying the L.H.S,
cos 2 2 x − cos 2 6 x = ( cos 2 x + cos 6 x ) ⋅ ( cos 2 x − cos 6 x ) = [ 2 cos ( 2 x + 6 x 2 ) ⋅ cos ( 2 x − 6 x 2 ) ] [ − 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 ) ] = ( 2 sin 4 x ⋅ cos 4 x ) ⋅ ( 2 sin 2 x ⋅ cos 2 x ) = sin 8 x ⋅ sin 4 x [ ∵ 2 sin θ cos θ = sin 2 θ ] = R . H . S
L.H.S. = R.H.S.
Hence, proved.
L.H.S.=
Using the trigonometric identities
Simplifying the L.H.S,
L.H.S. = R.H.S.
Hence, proved.
