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If n is integer then show that (1 + i...

If n is integer then show that
`(1 + i)^(2n) + (1 - i)^(2n) = 2 ^(n+1) cos . (npi)/2 ` .

Answer

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If n is a positive integer, show that (1 + i)^(n) + (1 - i)^(n) = 2 ^((n+2)/2) cos ((npi)/4) .

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Knowledge Check

  • 1 + i^(2) + i^(4) + i^(6) + … +i^(2n) =

    A
    positive
    B
    negative
    C
    0
    D
    cannot be determined

  • The value of ((1 + i)^(2n + 1))/((1- i)^(2n -1)) =

    A
    1
    B
    2
    C
    `pm 2`
    D
    `-2`

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    A : (1+i)^(6)+(1-i)^(6)=0 R : If n is a positive integer then (1+i)^(n)+(1-i)^(n)=2^((n//2)+1).cos""(npi)/(4)

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    If ( 1 + x ) ^(n) = a_(0) + a _(1) x + a_(2) x^(2) + ………. + a_(n)x^(n) , then show that (i) a_(0) - a_(2) + a_(4) - a_(6) + ……… = 2^(n//2) cos . (npi)/4 (ii) a _(1) - a_(3) + a_(5) - a_(7) + …….. = 2^(n//2) sin . (npi)/4

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