Question

If x+ 1/x = 2cos m then find x^n + 1/x^n ???????

Answer 1

Hey there !!!!!!!!!!!!!

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Let x be a complex number 

 x=cos(m)+isin(m)

Then 1/x will be conjugate of x i.e, cosm-isinm

So by using Demoivres theorem x =cosm+isinm=cism

                                                      1/x=cism-isinm=cis(-m)
cisⁿθ=cisnθ

So

xⁿ+1/xⁿ=cisⁿm+cisⁿ(-m)

           =cisnm+cis(-nm)

           =cos(nm)+isin(nm)+cos(-nm)+isin(-nm)

But cos(-∅)=cos∅ and sin(-∅)=-sin∅

So,

    cos(-nm)=cos(nm) and sin(-nm)=-sin(nm)

 =cos(nm)+isin(nm)+cos(nm)-isin(nm)

   xⁿ+1/xⁿ=2cos(nm)

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Hope this helped you..............