Question Number 151361 by mathdanisur last updated on 20/Aug/21
$$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}=\mathrm{45}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{x}\right)}{\mathrm{1}\:-\:\mathrm{tan}\left(\mathrm{y}\right)}\:=\:\mathrm{tan}\left(\mathrm{2y}\right) \\ $$
Answered by Ar Brandon last updated on 20/Aug/21
$${x}+{y}=\frac{\pi}{\mathrm{4}} \\ $$$$\frac{\mathrm{1}−\mathrm{tan}{x}}{\mathrm{1}−\mathrm{tan}{y}}=\frac{\mathrm{1}−\mathrm{tan}\left(\frac{\pi}{\mathrm{4}}−{y}\right)}{\mathrm{1}−\mathrm{tan}{y}} \\ $$$$=\frac{\mathrm{1}−\frac{\mathrm{1}−\mathrm{tan}{y}}{\mathrm{1}+\mathrm{tan}{y}}}{\mathrm{1}−\mathrm{tan}{y}}=\frac{\mathrm{2tan}{y}}{\mathrm{1}−\mathrm{tan}^{\mathrm{2}} {y}} \\ $$$$=\mathrm{tan2}{y} \\ $$
Commented by mathdanisur last updated on 20/Aug/21
$$\mathrm{Thank}\:\mathrm{You}\:\boldsymbol{\mathrm{S}}\mathrm{er} \\ $$