Published by:
CGP EDU Academic Team
Published on: August 13, 2026
If cos α + cos β + cos γ = sin α + sin β + sin γ prove that cos3 α + cos3 β + cos3 γ = 3cos( α + β + γ )
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
Sol. Let a = cos α + i sin α , b = cos β + i sin β and
c = cos γ + i sin γ
Then (a + b + c) = (cos α + cos β + cos γ ) +
i (sin α + sin β + sin γ ) = 0 + i 0 = 0 since a + b + c = 0
∴ a 3 + b 3 + c 3 = 3abc
⇒ (cos α + i sin α ) 3 + (cos β + i sin β ) 3 + (cos γ + i sin γ ) 3 = 3(cos α + i sin α ) (cos β + i sin β ) . (cos γ + i sin γ )
⇒ cos3 α + cos3 β + cos3 γ = 3cos ( α + β + γ )
(equating real and imaginary parts)
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