Cosx.cos2x. Cos4x.....cos(2*(n-1)x) =sin2(*n)x/2*n sinx prove using pmi

Since the identity is valid for n=1,cosx=sin2x2sinx=2sinx cosx2sinx=cosxNow, let the identity is true for n=m,cosx cos2x ....... cos2m-1x=sin 2mx2m sinxMultiplying both sides of above identity by cos2mx,cosx cos2x ....... cos2m-1x cos2mx=sin 2mx2m sinxcos2mx                                                                  =2sin 2mx cos2mx2m+1 sinx                                                                  =sin 2m+1x 2m+1 sinxThus, the identity holds for n=m+1Hence, the identity is proved for all positive integers n by mathematical induction.

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