Assuming the validity of the operations on the r.h.s. find `dy/dt`
`y = sqrt[[sin x + sqrt{{sin x + sqrt((sin x + ........)}])}]`

Explanation

`y = {sqrt [sin + sqrt {sin x +1........]}}`
sqrt(sin x + y)
implies y^2 + sin x + y implies 2y (dy)/(dx) = cos x + dy/dx
implies (2y-1) dy/dx = cos x implies dy/dx = cos x/(2y-1)`