Remember that putting a negative sign in front of a function means the same thing as multiplying that function by -1.
Let g(x) = -x2. Then we could think of this function as
g(x) = (-1)(x2),
g'(x) = (-1)(x2)' = (-1)(2x) = -2x.
The moral of the story is that the derivative of the negative of f is the negative of the derivative of f:
(-f(x))' = -f'(x).
Let f(x) = -sin x. Then
f'(x) = -(sin x)' = -cos(x).