Those aren't definitions; they're just facts about how $\cos$ and $\sin$ behave in the second quadrant. Those features follow from how the trig functions are defined using the unit circle: If $(x,y)$ is the point on the unit circle obtained by rotating counterclockwise by $\theta$ radians from the point $(1, 0)$, then $\cos(\theta) := x$ and $\sin(\theta) := y$.
These definitions are built specifically to agree with the "SOH CAH TOA" definitions for $\theta$ between $0$ and $\pi/2$; because the hypotenuse is 1, by definition, the sine of an angle is simply equal to the opposite side, which is $y$, and the cosine of an angle is equal to the adjacent side, which is $x$.
For $\theta$ in the interval from $\pi/2$ to $\pi$ (i.e., $\theta$ an obtuse angle), $(x, y)$ is in the second quadrant, where $x<0$ and $y>0$. Reflecting across the $y$-axis corresponds to taking the supplementary angle, and this reflection negates $x$ and leaves $y$ unchanged. Hence the cosine of $\theta$ is given by the negative of the cosine of the supplementary angle to $\theta$, and the sine is equal to the sine of the supplementary angle.
Since the supplementary angle is given by $\pi-\theta$, these two facts can be summarized by the equations $$\cos(\pi-\theta) = -\cos(\theta)$$ and $$\sin(\pi-\theta)=\sin(\theta),$$ both of which are special cases of the more general angle addition and subtraction formulas.