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Friday, April 5, 2024

2024/024) If \cos\, A +\cos\, B + \cos\, C = 0, prove that \cos 3A + \cos 3B +\cos 3C = 12 \cos\, A \cos\, B \cos\, C.

 We shall use 2 facts

\cos 3x = 4 \cos^3 x - 3 \cos\, x\cdots(1)

And 

if a+b+c=0 then a^3+b^3+c^3 = 3abc\cdots(2)

We are given

 \cos\, A +\ cos\, B + \cos\, C = 0\cdots(3)

Hence \cos^3 A +\ cos^3 B + \cos^3 C = 3 \cos\, A \cos\, B \cos\, C\cdots(4)

Now \cos 3A + \cos 3B +\cos 3C

= 4 \cos^3 A - 3 \cos\, A + 4 \cos^3 B - 3 \cos\, B +   4 \cos^3 C - 3 \cos\, C using(1)

=4(\cos^3 A +\ cos^3 B + \cos^3 C - 3 (\cos\, A + \cos\, B + \cos\, C)

=4 * 3 \cos\, A \cos\, B \cos\, C- 3 *0   using (3) and (4)

=12 \cos\, A \cos\, B \cos\, C

Proved


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