1. **State the problem:** Simplify and verify the expression $$(\sin x + \csc x)^2 + (\cos x + \sec x)^2 - \tan^2 x + \cot^2 x = 5.$$\n\n2. **Recall formulas and identities:**\n- $\csc x = \frac{1}{\sin x}$\n- $\sec x = \frac{1}{\cos x}$\n- $\tan^2 x + 1 = \sec^2 x$\n- $\cot^2 x + 1 = \csc^2 x$\n- $\cot x = \frac{1}{\tan x}$\n\n3. **Expand the squares:**\n$$(\sin x + \csc x)^2 = \sin^2 x + 2 + \csc^2 x$$\nExplanation: $\sin x \cdot \csc x = 1$, so $2 \sin x \csc x = 2$.\nSimilarly, $$(\cos x + \sec x)^2 = \cos^2 x + 2 + \sec^2 x.$$\n\n4. **Sum the expanded terms:**\n$$\sin^2 x + 2 + \csc^2 x + \cos^2 x + 2 + \sec^2 x = (\sin^2 x + \cos^2 x) + (\csc^2 x + \sec^2 x) + 4.$$\nSince $\sin^2 x + \cos^2 x = 1$, this becomes\n$$1 + \csc^2 x + \sec^2 x + 4 = 5 + \csc^2 x + \sec^2 x.$$\n\n5. **Simplify the remaining terms:**\nWe have to subtract $\tan^2 x$ and add $\cot^2 x$: $$5 + \csc^2 x + \sec^2 x - \tan^2 x + \cot^2 x.$$\n\n6. **Use Pythagorean identities:**\nFrom $\tan^2 x + 1 = \sec^2 x$, we get $\sec^2 x - \tan^2 x = 1$.\nFrom $\cot^2 x + 1 = \csc^2 x$, we get $\csc^2 x - \cot^2 x = 1$.\n\n7. **Rewrite the expression using these:**\n$$5 + \csc^2 x + \sec^2 x - \tan^2 x + \cot^2 x = 5 + (\csc^2 x - \cot^2 x) + (\sec^2 x - \tan^2 x) = 5 + 1 + 1 = 7.$$\n\n8. **Check the original problem statement:**\nThe original expression is $$(\sin x + \csc x)^2 + (\cos x + \sec x)^2 - \tan^2 x + \cot^2 x = 5.$$\nOur simplification shows it equals 7, not 5.\n\n9. **Re-examine the problem:**\nIf the problem intended $- \tan^2 x - \cot^2 x$ instead of $- \tan^2 x + \cot^2 x$, then\n$$5 + \csc^2 x + \sec^2 x - \tan^2 x - \cot^2 x = 5 + (\csc^2 x - \cot^2 x) + (\sec^2 x - \tan^2 x) - 2 \cot^2 x = 5 + 1 + 1 - 2 \cot^2 x,$$\nwhich is not simpler.\n\n10. **Conclusion:**\nThe expression simplifies to 7, not 5, so the given equation is incorrect as stated.\n\n**Final answer:** $$ (\sin x + \csc x)^2 + (\cos x + \sec x)^2 - \tan^2 x + \cot^2 x = 7.$$