The multiplicative inverse of $\frac{76}{91}$ is $\frac{91}{76}$. The multiplicative inverse of $\frac{65}{57}$ is $\frac{57}{65}$. The product of these inverses is $P = \frac{91}{76} \times \frac{57}{65}$. We can simplify this expression: $P = \frac{7 \times 13}{4 \times 19} \times \frac{3 \times 19}{5 \times 13}$. Cancelling out common factors ($13$ and $19$), we get $P = \frac{7 \times 3}{4 \times 5} = \frac{21}{20}$. The additive inverse of $P$ is $-P = -\frac{21}{20}$.