Prove that cos &#x2061;<!-- ⁡ --> ( 2 n x ) = <msubsup> &#x2211;

William Santiago 2022-05-15 Answered
Prove that cos ( 2 n x ) = k = 0 n ( 1 ) k ( 2 n 2 k ) cos 2 ( n k ) ( x ) sin 2 k ( x ) := p ( n )
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Answers (1)

Calvin Oneill
Answered 2022-05-16 Author has 20 answers
Not by induction:
cos 2 n x = [ cos 2 n x + i sin 2 n x ] = [ ( cos x + i sin x ) 2 n ] = [ j = 0 2 n ( 2 n j ) cos 2 n j x i j sin j x ] = k = 0 n ( 2 n 2 k ) cos 2 n 2 k x i 2 k sin 2 k x ( j = 2 k ) = k = 0 n ( 1 ) k ( 2 n 2 k ) cos 2 ( n k ) x sin 2 k x
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