Question

Find it using de moivres theorem

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

Answer 1

Hi friend,
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=cis(-m)
cisⁿθ=cisnθ
So xⁿ+1/xⁿ=cisⁿm+cisⁿ(-m)
                 =cisnm+cis(-nm)
                 =cos(nm)+isin(nm)+cos(-nm)+isin(-nm)
               cos(nm)=cos(-nm) and sin(-nm)=-sin(nm)
 =cos(nm)+isin(nm)+cos(nm)-isin(nm)
                =2cos(nm)
Hope this helps..............