Let A and B denote the statements A: cos α + cos β + cos γ = 0 B: sin α + sin β + sin γ = 0 If cos ( β − γ ) + cos ( γ − α ) + cos ( α − β ) = − 2 3 , then

Question

Let A and B denote the statements A: cos α + cos β + cos γ = 0 B: sin α + sin β + sin γ = 0 If cos ( β − γ ) + cos ( γ − α ) + cos ( α − β ) = − 2 3 ​ , then, A is true and B is false, A is false and B is true, both A and B are true, both A and B are false

MARKS App · Accepted Answer

$ ​ cos ( β − γ ) + cos ( γ − α ) + cos ( α − β ) = − 2 3 ​ ⇒ 2 [ cos ( β − γ ) + cos ( γ − α ) + cos ( α − β )] + 3 = 0 ⇒ 2 [ cos ( β − γ ) + cos ( γ − α ) + cos ( α − β )] + sin 2 α + cos 2 α + sin 2 β + cos 2 β + sin 2 γ + cos 2 γ = 0 ⇒ ( sin α + sin β + sin γ ) 2 + ( cos α + cos β + cos γ ) 2 = 0 ​ $

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