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For a positive integer n show that (1+is...

For a positive integer n show that `(1+isqrt3)^n+(1-isqrt3)^n=2^(n+1) "cos"(npi)/3`

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`L.H.S.=(1+isqrt3)^n+(1-isqrt3)^n`
`{2(1/2+isqrt3/2)}^n+{2(1/2-isqrt3/2)}^4`
`=2^n{("cos"pi/3="isin"pi/3)^n+("cos"pi/3-"isin"pi/3)}^n`
`=2^n("cos"(npi)/3+"isin"(npi)/3+"cos"(npi)/3-"sin"(npi)/3)`
`=2^n 2"cos"(npi)/3=2^(n+1)"cos"(npi)/3=R.H.S`
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