Given, ${S_1} = \left\{ {x \in \left[ {0,12\pi } \right],\,{{\sin }^5}x + {{\cos }^5}x = 1} \right\}$ ${S_2} = \left\{ {x \in \left[ {0,8\pi } \right],\,{{\sin }^7}x + {{\cos }^7}x = 1} \right\}$ (1) ${\sin ^5}x + {\cos ^5}x = 1$ This satisfies when $\sin x = 1$ and $\cos x = 0$ $\therefore$ $x = {\pi \over 2},{{5\pi } \over 2},{{9\pi } \over 2},{{13\pi } \over 2},{{17\pi } \over 2},{{21\pi } \over 2}$ It also satisfies when $\sin x = 0$ and $\cos x = 1$ $\therefore$ $x = 0,\,2\pi ,4\pi ,6\pi ,8\pi ,10\pi ,12\pi$ $\therefore$ Accepted values of x in $\left[ {0,12\pi } \right]$ is = 13 $\therefore$ $n({S_1}) = 13$ (2) ${\sin ^7}x + {\cos ^7}x = 1$ This satisfies when $\sin x = 1$ and $\cos x = 0$ For $x \in \left[ {0,\,8\pi } \right]$, possible values $x = {\pi \over 2},{{5\pi } \over 2},{{9\pi } \over 2},{{13\pi } \over 2}$ It also satisfies when $\sin x = 0$ and $\cos x = 1$ $x \in \left[ {0,8\pi } \right]$, possible values $x = 0,2\pi ,4\pi ,6\pi ,8\pi$ $\therefore$ Total accepted values of x in $\left[ {0,8\pi } \right]$ is = 9 $\therefore$ $n({S_2}) = 9$ $\therefore$ $n({S_1}) - n({S_2}) = 13 - 9 = 4$