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Kyla Ayers

Kyla Ayers

Answered question

2022-06-24

Prove that
1 + ( n 1 ) cos x + ( n 2 ) cos 2 x + . . . cos n x = ( 2 cos x 2 ) n ( cos n x 2 )
given that
( 1 + cos x + i sin x ) n = ( 2 cos x 2 ) n ( cos n x 2 + i sin n x 2 )
I'm not so sure how to even start with this, any hints?

Answer & Explanation

Dustin Durham

Dustin Durham

Beginner2022-06-25Added 31 answers

Hint: We have
k = 0 n ( n k ) e k i x = ( 1 + e i x ) n = ( 1 + cos ( x ) + i sin ( x ) ) n
then
k = 0 n ( n k ) cos ( k x ) = Re ( k = 0 n ( n k ) e k i x ) = Re ( ( 1 + e i x ) n )
now use the given identity.

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