Question Number 141786 by Rankut last updated on 23/May/21
$$\boldsymbol{\mathrm{If}}\:\:\boldsymbol{\mathrm{y}}=\mathrm{1}−\boldsymbol{\mathrm{cos}}\mathrm{2}\boldsymbol{\mathrm{t}}\:\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{x}}=\sqrt{\mathrm{1}+\boldsymbol{\mathrm{t}}^{\mathrm{2}} }\:.\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{\mathrm{dy}}}{\boldsymbol{\mathrm{dx}}}=\frac{\mathrm{2}\sqrt{\mathrm{1}+\boldsymbol{\mathrm{t}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{sin}}\mathrm{2}\boldsymbol{\mathrm{t}}}{\boldsymbol{\mathrm{t}}} \\ $$$$\boldsymbol{\mathrm{Help}}\:\boldsymbol{\mathrm{please}}. \\ $$
Answered by iloveisrael last updated on 23/May/21
$$\left(\mathrm{1}\right)\:\frac{{dx}}{{dt}}\:=\:\frac{{t}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:;\:\frac{{dt}}{{dx}}\:=\:\frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{{t}} \\ $$$$\left(\mathrm{2}\right)\:\frac{{dy}}{{dt}}\:=\:\mathrm{2sin}\:\mathrm{2}{t} \\ $$$$\left(\mathrm{3}\right)\:\frac{{dy}}{{dx}}\:=\:\frac{{dy}}{{dt}}\:.\:\frac{{dt}}{{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{2sin}\:\mathrm{2}{t}\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{{t}} \\ $$