Question Number 117568 by ZiYangLee last updated on 12/Oct/20
$$\mathrm{Suppose}\:\mathrm{the}\:\mathrm{non}-\mathrm{constant}\:\mathrm{functions}\:{f}\:\mathrm{and}\:{g} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{following}\:\mathrm{two}\:\mathrm{conditions}: \\ $$$$\mathrm{I}:\:{g}\left({x}−{y}\right)={g}\left({x}\right){g}\left({y}\right)+{f}\left({x}\right){f}\left({y}\right)\:\forall\:{x},{y}\in\mathbb{R} \\ $$$$\mathrm{II}:\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\mathrm{Evaluate} \\ $$$$\mathrm{i}.\:{g}\left(\mathrm{0}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{ii}.\left[{f}\left({x}\right)\right]^{\mathrm{2}} +\left[{g}\left({x}\right)\right]^{\mathrm{2}} \\ $$
Commented by prakash jain last updated on 13/Oct/20
$${g}\left({x}\right)=\mathrm{cos}\:\left({x}\right) \\ $$$${f}\left({x}\right)=\mathrm{sin}\:\left({x}\right) \\ $$
Answered by Olaf last updated on 12/Oct/20
$$ \\ $$$$\mathrm{If}\:{x}\:=\:{y} \\ $$$${g}\left(\mathrm{0}\right)\:=\:\left[{f}\left({x}\right)\right]^{\mathrm{2}} +\left[{g}\left({x}\right)\right]^{\mathrm{2}} \:\left(\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{If}\:{x}\:=\:{y}\:=\:\mathrm{0} \\ $$$${g}\left(\mathrm{0}\right)\:=\:\left[{f}\left(\mathrm{0}\right)\right]^{\mathrm{2}} +\left[{g}\left(\mathrm{0}\right)\right]^{\mathrm{2}} \:=\:\left[{g}\left(\mathrm{0}\right)\right]^{\mathrm{2}} \\ $$$$\Rightarrow\:{g}\left(\mathrm{0}\right)\:=\:\mathrm{0}\:\mathrm{or}\:{g}\left(\mathrm{0}\right)\:=\:\mathrm{1} \\ $$$$ \\ $$$$\mathrm{But}\:\mathrm{if}\:{g}\left(\mathrm{0}\right)\:=\:\mathrm{0},\:\left(\mathrm{1}\right)\:\Rightarrow\:{f}\left({x}\right)\:=\:{g}\left({x}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{Then}\:{f}\:\mathrm{and}\:{g}\:\mathrm{are}\:\mathrm{constant}\:\mathrm{functions}. \\ $$$$\mathrm{Impossible}\:! \\ $$$$ \\ $$$$\mathrm{Finally}\:{g}\left(\mathrm{0}\right)\:=\:\mathrm{1}\:\mathrm{and}\:: \\ $$$$\left[{f}\left({x}\right)\right]^{\mathrm{2}} +\left[{g}\left({x}\right)\right]^{\mathrm{2}} \:=\:\mathrm{1} \\ $$
Commented by ZiYangLee last updated on 12/Oct/20
$$\mathrm{Good}!! \\ $$