sin and cos

x=cost , y=sint,find (d^2 y)/ (d x^2),
Sol
dy/dx
=(dy/dt)*(dt/dx)
=(dy/dt)*[dx/dt)]^(－1)
=(dsint/dt)*[dcost/dx]^(－1)
=cost*[-sint]^(－1)
=－cott
so
(d^2y)/(dx^2)
=－dcott/dx
=－dcott/dt*(dt/dx)
=csc^2t*(dx/dt)^(－1)
=csc^2t*(dcost/dt)^(－1)
=csc^2t*(－sint)^*(－1)
=－csc^2t*csct
=－csc^3t
=－1/sin^3t

x^2 + y^2 = cos^2 t + sin^2 t = 1
2x + 2y(dy/dx) = 0
dy/dx = -x/y
d^2 y/dx^2 = [-y + x (dy/dx)]/y^2
= [-y + x(-x/y)]/y^2
= [-y - x^2/y]/y^2
= -(y^2 + x^2)/y^3
= -1/y^3
= -1/sin ^3 t.