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If-y-1-cos2t-and-x-1-t-2-Show-that-dy-dx-2-1-t-2-sin2t-t-Help-please-




Question Number 141786 by Rankut last updated on 23/May/21
If  y=1−cos2t  and x=(√(1+t^2 )) .Show that  (dy/dx)=((2(√(1+t^2 )) sin2t)/t)  Help please.
$$\boldsymbol{\mathrm{If}}\:\:\boldsymbol{\mathrm{y}}=\mathrm{1}−\boldsymbol{\mathrm{cos}}\mathrm{2}\boldsymbol{\mathrm{t}}\:\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{x}}=\sqrt{\mathrm{1}+\boldsymbol{\mathrm{t}}^{\mathrm{2}} }\:.\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{\mathrm{dy}}}{\boldsymbol{\mathrm{dx}}}=\frac{\mathrm{2}\sqrt{\mathrm{1}+\boldsymbol{\mathrm{t}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{sin}}\mathrm{2}\boldsymbol{\mathrm{t}}}{\boldsymbol{\mathrm{t}}} \\ $$$$\boldsymbol{\mathrm{Help}}\:\boldsymbol{\mathrm{please}}. \\ $$
Answered by iloveisrael last updated on 23/May/21
(1) (dx/dt) = (t/( (√(1+t^2 )))) ; (dt/dx) = ((√(1+t^2 ))/t)  (2) (dy/dt) = 2sin 2t  (3) (dy/dx) = (dy/dt) . (dt/dx)                 = ((2sin 2t (√(1+t^2 )))/t)
$$\left(\mathrm{1}\right)\:\frac{{dx}}{{dt}}\:=\:\frac{{t}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:;\:\frac{{dt}}{{dx}}\:=\:\frac{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{{t}} \\ $$$$\left(\mathrm{2}\right)\:\frac{{dy}}{{dt}}\:=\:\mathrm{2sin}\:\mathrm{2}{t} \\ $$$$\left(\mathrm{3}\right)\:\frac{{dy}}{{dx}}\:=\:\frac{{dy}}{{dt}}\:.\:\frac{{dt}}{{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{2sin}\:\mathrm{2}{t}\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{{t}} \\ $$

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