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If x = a sin t and y = a (cost+log tan(t/2)) ,find ((d^2y)/(dx^2)) - Mathematics and Statistics

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If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`

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Solution

`y = a (cost+logtan(t/2))  and x=asint`

`therefore dy/dt=a[d/dt(cost)+d/dt(log tan (t/2))]=a[-sint+cot(t/2)xxsec^2(t/2)xxt/2]=a[-sint+1/(2sin(t/2)cos(t/2))]`

`dy/dt=a(-sint+1/sint)=a((-sin^2t+1)/sint)=a cos^2t/sint`

`dx/dt=a d/dt(sint)=acost`

`therefore dy/dx=(dy/dt)/(dx/dt)=(a (cos^2t/sint))/acost=cost/sint=cott`

`(d^2y)/(dx^2)=d(cott)/dx=-cosec^2tdt/dx=-cosec^2txx1/(acost)=1/(asin^2tcost)`

 

 

Concept: Second Order Derivative
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