Question Number 50908 by Smail last updated on 22/Dec/18
$${Given}\:{f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} {sin}\left({kx}\right){cos}\left(\left({n}−{k}\right){x}\right) \\ $$$${Find}\:{a}\:{simple}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left({Your}\:{answer}\:{should}\:{be}\:{written}\:{like}\:{c}\left({n}\right).{g}\left({nx}\right)\right)\: \\ $$
Answered by Smail last updated on 23/Dec/18
$${f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} {sin}\left({kx}\right){cos}\left(\left({n}−{k}\right){x}\right) \\ $$$${let}\:{l}={n}−{k} \\ $$$${f}\left({x}\right)=\underset{{l}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{n}−{l}} {sin}\left(\left({n}−{l}\right){x}\right){cos}\left({lx}\right) \\ $$$$=\underset{{l}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{l}} {sin}\left(\left({n}−{l}\right){x}\right){cos}\left({lx}\right) \\ $$$$\mathrm{2}{f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} \left({sin}\left(\left({n}−{k}\right){x}\right){cos}\left({kx}\right)+{sin}\left({kx}\right){cos}\left(\left({n}−{k}\right){x}\right)\right) \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} \left({sin}\left(\left({n}−{k}\right){x}+{kx}\right)\right) \\ $$$$\mathrm{2}{f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} {sin}\left({nx}\right)=\mathrm{2}^{{n}} {sin}\left({nx}\right) \\ $$$${f}\left({x}\right)=\mathrm{2}^{{n}−\mathrm{1}} {sin}\left({nx}\right) \\ $$