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Mohamed Mooney

Mohamed Mooney

Answered question

2022-06-13

Prove: cos 3 A + cos 3 ( 120 ° + A ) + cos 3 ( 240 ° + A ) = 3 4 cos 3 A

Answer & Explanation

Lilliana Burton

Lilliana Burton

Beginner2022-06-14Added 19 answers

You can show that if
a+b+c=0
then
a 3 + b 3 + c 3 = 3 a b c
Since, in your problem, for every A
cos A + cos ( A + 2 π / 3 ) + cos ( A 2 π / 3 ) = 0
Then you can use the above identity
cos 3 A + cos 3 ( A + 2 π / 3 ) + cos 3 ( A 2 π / 3 ) = 3 cos A cos ( A + 2 π / 3 ) cos ( A 2 π / 3 )
By using the previously mentioned identity
cos x cos y = 1 2 ( cos ( x + y ) + cos ( x y ) )
you can simplify the RHS to this
3 2 cos A ( cos ( 2 A ) 1 2 )
and then (using it again)
3 4 ( cos 3 A + cos ( A ) ) 3 4 cos A = 3 4 cos ( 3 A )

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