Solution: Expand the expression: $(\sin x + \csc x)^2 = \sin^2 x + 2 + \csc^2 x$ and $(\cos x + \sec x)^2 = \cos^2 x + 2 + \sec^2 x$. Combine terms: $\sin^2 x + \cos^2 x + 4 + \csc^2 x + \sec^2 x$. Since $\sin^2 x + \cos^2 x = 1$, this simplifies to $5 + \csc^2 x + \sec^2 x$. Rewrite $\csc^2 x = 1 + \cot^2 x$ and $\sec^2 x = 1 + \tan^2 x$, so total becomes $7 + \tan^2 x + \cot^2 x$. Let $t = \tan^2 x$, then expression is $7 + t + \frac1t$. The minimum of $t + \frac1t$ for $t > 0$ is $2$ by AM-GM inequality. Thus, the minimum value is $7 + 2 = \boxed9$. - Abu Waleed Tea