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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 $7 + 2 = \boxed{9}$. 📰 You Wont Believe How Lewis Structure For Ch Changes Science Homework Forever 2380066 📰 How Long Does Donating Plasma Take 4008168 📰 Packing Life 6759556 📰 How Long Is 20 Weeks 5935819 📰 Unlock The Power Of The 1111 Angel Number What Happens When It Appears 11 Times 1970761 📰 Brazosport College 892344 📰 Npi Search For Md Uncovered These Surprising Results Will Shock You 3969064 📰 H Funding Grab Alert Get Massive Capital Fastexclusive Insider Tips Inside 4497550 📰 El Juntos Seguros Que Transforma Vidaslo Que Nunca Dejaron En La Trueza 3776043 📰 Cast Lord Of The Rings 3 8151775 📰 The Secret Spring Wallpaper That Transforms Your Walls Into A Fresh Floral Escapewatch Your Home Ignite 614780 📰 Unlock Cherry Magics Magicgrant Yourself Eternal Bliss And Wonder 128570 📰 These Bathroom Sconces Will Transform Your Spaceyou Wont Believe How Easy They Are To Install 4151103 📰 Find The Area Enclosed By The Curves Y X2 And Y 4X X2 1366972 📰 Kerri Strug 9050417