The number of solutions of

$\sin ^{7} x \leq \sin ^{2} x \leq 1$ ..........(1)

and $\cos ^{7} x \leq \cos ^{2} x \leq 1$ ...........(2)

also $\sin ^{2} x+\cos ^{2} x=1$

$\Rightarrow$ equality must hold for (1) \& (2)

$\Rightarrow \sin ^{7} x=\sin ^{2} x \& \cos ^{7}=\cos ^{2} x$

$\Rightarrow \sin x=0 \& \cos x=1$

or

$\cos x=0 \& \sin x=1$

$\Rightarrow x=0,2 \pi, 4 \pi, \frac{\pi}{2}, \frac{5 \pi}{2}$

$\Rightarrow 5$ solutions