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