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

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

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

$\Rightarrow \text { equality must hold for }(1) \,\,(2)$

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

$\Rightarrow \sin x=0\, \, \cos x=1 \text { or }$

$\cos x=0\,  \,\sin x=1$

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

$\Rightarrow 5 \text { solutions }$