I. (a) cos 24 + cos 2B + cos 2C=-1-4 cos A cos B cos C.

1.	I. (a) cos 24 + cos 2B + cos 2C=-1-4 cos A cos B cos C.
Answer» The result should be : If A + B + C = π, then cos 2A + cos 2B + cos 2C = -1 - 4 cos A cos B cos C._______________________________ ... LHS = ( cos 2A + cos 2B ) + cos 2C = [ 2 cos (A+B) cos (A-B) ] + cos 2C = [ 2 cos (π-C) cos (A-B) ] + ( 2 cos² C - 1 ) = [ 2( -cos C ) cos (A-B) ] + 2 cos² C - 1 = -1 - 2 cos C cos (A-B) + 2 cos² C = -1 - 2 cos C [ cos (A-B) - cos C ] = -1 - 2 cos C [ cos (A-B) - ( -cos (A+B)) ] = -1 - 2 cos C [ cos (A+B) + cos (A-B) ] = -1 - 2 cos C [ 2 cos A cos B ] = -1 - 4 cos A cos B cos C = RHS ........................................... Q.E.D.

Answer»

The result should be : If A + B + C = π, then

cos 2A + cos 2B + cos 2C = -1 - 4 cos A cos B cos C._______________________________

... LHS

= ( cos 2A + cos 2B ) + cos 2C

= [ 2 cos (A+B) cos (A-B) ] + cos 2C

= [ 2 cos (π-C) cos (A-B) ] + ( 2 cos² C - 1 )

= [ 2( -cos C ) cos (A-B) ] + 2 cos² C - 1

= -1 - 2 cos C cos (A-B) + 2 cos² C

= -1 - 2 cos C [ cos (A-B) - cos C ]

= -1 - 2 cos C [ cos (A-B) - ( -cos (A+B)) ]

= -1 - 2 cos C [ cos (A+B) + cos (A-B) ]

= -1 - 2 cos C [ 2 cos A cos B ]

= -1 - 4 cos A cos B cos C

= RHS ........................................... Q.E.D.

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