What is the general solution of (dy/dx)sqrt(1-x^2) + sqrt(1-y^2) = 0

The problem gives a hint to use the addition formula for the sine function.

Please show your steps! Thank you in advance.

Separable:
dy / √(1 - y²) = -dx / √(1 - x²)

Integrating (you do remember your inverse trig derivatives right?
arcsin(y) = -arcsin(x) + C

Then taking the sine of both sides gives
y = sin(C - arcsin(x))

and this is where you'd use the sum of angles identity to pretty it up.