Question Number 102342 by bemath last updated on 08/Jul/20

If { ((x=2t+sin 2t)),((y=e^(sin 2t) )) :} prove that (1/y).(dy/dx) = tan ((π/4)−t)

Commented byDwaipayan Shikari last updated on 08/Jul/20

There is some error in question if x=2t−cos2t then the prove is true

Commented bybobhans last updated on 08/Jul/20

yes sir. i think something error in question

Answered by bobhans last updated on 08/Jul/20

⇒sin 2t = ln(y) ; 2 cos 2t = (1/y).(dy/dt) (dx/dt) = 2+2cos 2t ⇒(dx/dt) = 2+(1/y).(dy/dt) ...(1) (dy/dx) = (dy/dt)×(dt/dx) = (dy/dt)×(1/(dx/dt)) (dy/dx)= 2y cos 2t × (1/(2+2cos 2t)) (1/y) (dy/dx) = ((2cos 2t)/(2(1+cos 2t))) = ((1−2sin ^2 t)/(2cos ^2 t)) =(1/2)sec ^2 t−tan ^2 t = (1/2)(tan ^2 t−1)−tan ^2 t

Answered by Dwaipayan Shikari last updated on 08/Jul/20

if x=2t−cos2t (dx/dt)=2+2sin2t logy=sin2t (1/y).(dy/dt)=2cos2t (1/y).(dy/dx)=((2cos2t)/(2+2sin2t))=(((1−tan^2 t)/(1+tan^2 t))/(1+((2tant)/(1+tan^2 t))))=((1−tan^2 t)/((1+tant)^2 ))=((1−tant)/(1+tant))=tan((π/4)−t)