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
No comments:
Post a Comment