Show that y=sin (t) is a solution to (dydt) 2=1-y2. Enter your answers below in terms of the independent variable t in the order in which the terms were given. Be sure you can justify your answer.

y = sin (t) is a solution to the differential equation

(dy/dt) ² = 1 - y²

Step-by-step explanation:

Given (dy/dt) ² = 1 - y²

Suppose y = sin (t) is a solution, then it satisfies the differential equation.

That is

[d (sin (t)) ]² = 1 - y²

Let y = sin (t)

dy/dt = d (sin (t)) = cos (t)

(dy/dt) ² = cos²t

But cos²t + sin²t = 1

=> 1 - sin²t = cos²t

So

(dy/dt) ² = 1 - sin²t

Since sin²t = (sint) ² = y²,

we have

(dy/dt) ² = 1 - y²

as required.