If `x^2=t^2+!` then find `(d^2x)/(dt^2)`

If x=acosn t-bsinn t and (d^2x)/(dt^2)=lambdax , then find the value of lambda .

If x=a (cos t + t sin t) and y = a (sin t - t cos t ) , find (d^2x)/(dt^2),(d^2y)/(dt^2) and (d^2y)/(dx^2) . Also mention the domain of validity.

If x= cos t+ log tan t/2 and y = sin t, then find the value of (d^2y)/(dt^2) and (d^2y)/(dx^2) at t= (pi)/4 .

If x=cos t + (log tant)/2, y=sin t, then find the value of (d^2y)/(dt^2) and (d^2y)/(dx^2)  at t =pi/4.

If x Aos 4t+Bsin4t then (d^(2)x)/(dt^(2)) =

If x=cost+logtan(t/2),\ \ y=sint , then find the value of (d^2y)/(dt^2) and (d^2y)/(dx^2) at t=pi/4 .

If x = a ( cos t + t sint) and y = a (sin t - t cost), 0 < t < pi/2 , find (d^2x)/(dt^2),(d^2y)/(dt^2)

If x = e ^(2t ) cos, 3t, then (d ^(2) x)/(dt ^(2))at t = pi //2 is

If x=cos t+(log tan t)/(2),y=sin t, then find the value of (d^(2)y)/(dt^(2)) and (d^(2)y)/(dx^(2)) at t=(pi)/(4)