Proving Angle Addition/Subtraction Formulas Practice Test

Which explanation correctly uses the geometry?

In triangle $ABC$, $\angle A$ is labeled $\theta$ and $\angle B$ is labeled $\varphi$. A segment from $C$ to $AB$ meets $AB$ at $D$ and is marked perpendicular to $AB$ (right-angle box at $D$). Thus $CD$ is an altitude.

Which statement gives a valid geometric justification path toward an identity involving $\sin(\theta+\varphi)$ (using that $\angle C = 180^\circ-(\theta+\varphi)$), without plugging in numbers or assuming the identity?

Which statement proves the formula?

A right triangle $\triangle OAB$ has right angle at $A$, with $OA$ along the positive $x$-axis and $AB$ vertical (right-angle box at $A$). The angle at $O$ between $OA$ and $OB$ is labeled $\theta+\varphi$. A ray from $O$ inside the triangle meets $AB$ at point $C$ and splits the angle at $O$ into $\angle AOC=\theta$ and $\angle COB=\varphi$.

Which reasoning correctly leads to an identity for $\tan(\theta+\varphi)$ using this decomposition?