Sine and Cosine functions possess notable formulas for addition of angles $$ \sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b). $$ One can come up with several $q$ analogs of trigonometric functions. For instance, from the relation $$ \sin(\pi z) = \frac{\pi}{\Gamma(z)\Gamma(1-z)}, $$ one can define the $q$-sine function as $$ \sin_q(\pi z) = \frac{\pi}{\Gamma_q(z)\Gamma_q(1-z)}. $$

My question is: Are there notable $q$ deformations of trigonometric functions satisfying addition formulas for angles?