Question

cos3a cos 3 b+ cos3b cos3c + cos3c cos3a= 3 if a+b+c=180​

Answer 1

Step-by-step explanation:

given, A + B + C = 180°

so, 3A + 3B + 3C = 540°

LHS = sin3A + sin3B + sin3C

= 2sin(3A + 3B)/2. cos(3A - 3B)/2 + sin3C

= 2sin(540° - 3C)/2 . cos(3A - 3B)/2 + sin3C

= 2sin(270° - 3C/2) . cos(3A - 3B)/2 + 2sin3C/2. cos3C/2

= -2cos3C/2 . cos(3A - 3B)/2 + 2cos3C/2 . sin3C/2

= -2cos3C/2 [ cos(3A - 3B)/2 - sin3C/2 ]

= -2cos3C/2 [ cos(3A - 3B)/2 + cos(270° - 3C/2)]

= -2cos3C/2 [ 2cos(3A/2- 3B/2 + 270° - 3C/2)/2. cos(3A/2 - 3B/2 - 270° + 3C/2)/2]

= -4cos3C/2 [ cos{3A/2 + 270° - (3B + 3C)/2}/2.cos{(3A + 3C)/2 - 3B/2 - 270°}/2 ]

= -4cos3C/2 [ cos(3A/2 + 3A/2)/2 . cos(- 3B/2 - 3B/2)/2 ]

= -4cos3A/2 . cos3B/2 cos3C/2 = RHS

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#ANSWER WITH QUALITY....

TQ.....