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📰 Final cases = $ 10 \times 2^5 = 10 \times 32 = 320 $. 📰 #### 320Question: Find the minimum value of $(\sin x + \csc x)^2 + (\cos x + \sec x)^2$. 📰 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 $7 + t + \frac{1}{t}$. The minimum of $t + \frac{1}{t}$ for $t > 0$ is $2$ by AM-GM inequality. Thus, the minimum value is $7 + 2 = \boxed{9}$. 📰 Best Fast Food Burger 7920376 📰 Credit Card With No Foreign Fee 8955961 📰 Vigo The Carpathian The Hidden Hero You Need To Know About In 2024 2294985 📰 Secrets Hidden In Instagram Storiesstories You Never Post Again 9708991 📰 How To Delete An Account On Uber 4573711 📰 The Shocking Truth About The Sonic Timeline That Will Change Everything 4813643 📰 Shockingly Hidden Net Worth Of Scott Boras How Much Is He Really Worth 694630 📰 Sydney Sweeney Silver Dress 4598273 📰 Count Only Unique Values In Excelreveal Hidden Data Secrets Now 8730553 📰 This Lausd Digital Classroom Hack Is Spreading Fastno One Can Ignore It Anymore 3567827 📰 Play Now Before Its Gone Fast Paced Games Youll Want To Keep Playing 4668630 📰 Iphone Cleaning App 137169 📰 Ms Whitman Lyrics 7793079 📰 5Revive Your Crops Daily With Ciproud The Revolutionary Plant Booster 746179 📰 Another Word For Vulnerability 6507970