JEE Previous Year Question

If $k=\text{tan}(\frac{\pi }{4}+\frac{1}{2}{\text{cos}}^{-1}\left(\frac{2}{3}\right))+\text{tan}\left(\frac{1}{2}{\text{sin}}^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation ${\text{sin}}^{-1}\text{(}\text{kx}-1\text{)}={\text{sin}}^{-1}x-{\text{cos}}^{-1}x$ is $\text{\_}\text{\_}\text{\_}\text{\_}$

Topic: JEE PYQ

Answer: 1. Full step-by-step solution below.