Determine whether the statement "I’ve noticed that for sine, cosine, and tangent, the trig function for the sum of two angles is not equal to that trig function of the first angle plus that trig function of the second angle", makes sense or does not make sense, and explain your reasoning.

The statement is “ The trigonometric function for the sum of two angles is not equal to the trigonometric function of the first angle plus trigonometry function of the second angle.” i.e. in a mathematical way, we can say
${\displaystyle \sin (A+B)\ne \sin \left(A\right)+\sin \left(B\right)}$
${\displaystyle \cos (A+B)\ne \cos \left(A\right)+\cos \left(B\right)}$
${\displaystyle \tan (A+B)\ne \tan \left(A\right)+\tan \left(B\right)}$
The statement is True.
Since the formulas for the trigonometric function of the sum of two angles are
${\displaystyle \sin (A+B)=\sin \left(A\right)\cos \left(B\right)+\cos \left(A\right)\sin \left(B\right)}$
${\displaystyle \cos (A+B)=\cos \left(A\right)\cos \left(B\right)+\sin \left(A\right)\sin \left(B\right)}$
${\displaystyle \tan (A+B)=\frac{\tan \left(A\right)+\tan \left(B\right)}{1-\tan \left(A\right)\tan \left(B\right)}}$
Hence generally the trigonometric function for the sum of two angles is not equal to the trigonometric function of the first angle plus trigonometry function of the second angle.