Question

If `x=a (cos t +t sint )and y= a(sint-cos t )`  Prove that `Sec^3 t/(at),0<t< pi/2` 

Answer 1

It is given that , `x=a (cos t+t sin t)and y=a(sin t-t cos t)`

`∴ dx/dt=a.d/dt (cos t+t sin t)`

`= a [-sin t + sin t. d/dt(t)+t.d/dt(sin t)]`

=`a[-sin t+sin t+t cos t]=at cos t`

`dy/dt=a. d/dt(sin t- cost t)`

`=a[cos t-{cos t.d/dt(t)+t. d/dt(cos t)}]` 

`=a[cos t-{cos t-t sin t}]=at sin t`

`∴ dy/dx=((dy/dt))/((dx/dt))=(at sin t)/(at cos t)=tan t`

Then, ` d^2 y/dx^2=d/dx (dy/dx)=d/dx(tan t)=sec^2 t. dt/dx`

`=sec^2 t. 1/(at cos t)   [dx/dt=at cost ⇒ dt/dx=1/(at cos t)]`

`= sec^3t/(at), 0<t< pi/2`