Distinguishing rotating Kiselev black hole from naked singularity using spin precession of test gyroscope

We study the critical values ($\alpha_{c}, a_{c}$) of the quintessential and spin parameters, to distinguish a rotating Kiselev black hole (RKBH) from a naked singularity. For any value of the dimensionless quintessential parameter $\omega_{q} \in (-1, -1/3)$, a black hole can exist if $\alpha$ $\leq$ $\alpha_{c}$ and $a$ $\leq$ $a_{c}$. Both $\alpha_{c}$ and $a_{c}$ are directly proportional to $\omega_{q}$. Further, for any $\omega_{q}$, when increasing the value of $\alpha$ from zero to $\alpha_{c}$, the size of the event horizon increases, whereas the size of the outer horizon decreases. We also study the critical value of the quintessential parameter $\overline{\alpha}_c$ for the Kiselev black hole (KBH) and find horizons of extremal black holes. It is seen that, similar to a RKBH, for a KBH with any $\omega_{q}$, when increasing the value of $\alpha$, the size of the event horizon increases, while the size of the outer horizon decreases. We then study the spin precession of a test gyroscope attached to a stationary observer in this spacetime. Using the spin precessions we differentiate black holes from naked singularities. If the precession frequency becomes arbitrarily large, as approaching to the central object in the quintessential field along any direction, then the spacetime is a black hole. A spacetime will contain a naked singularity if the precession frequency remains finite everywhere except at the singularity itself. Finally, we study the Lense-Thirring precession frequency for RKBHs and the geodetic precession for KBHs.