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📰 Question: Find the minimum value of $ (\cos x + \sec x)^2 + (\sin x + \csc x)^2 $. 📰 Solution: Expand the expression: $ \cos^2x + 2 + \sec^2x + \sin^2x + 2 + \csc^2x $. Simplify using $ \cos^2x + \sin^2x = 1 $ and $ \sec^2x = 1 + an^2x $, $ \csc^2x = 1 + \cot^2x $: $ 1 + 2 + 1 + an^2x + 1 + \cot^2x + 2 = 7 + an^2x + \cot^2x $. Let $ t = an^2x $, so $ \cot^2x = rac{1}{t} $. The expression becomes $ 7 + t + rac{1}{t} $. By AM-GM, $ t + rac{1}{t} \geq 2 $, so the minimum is $ 7 + 2 = 9 $. Thus, the minimum value is $ oxed{9} $. 📰 Question: Among all roots of $ z^6 + z^4 + z^2 + 1 = 0 $, the maximum imaginary part can be expressed as $ \sin heta $, where $ 0 < heta < rac{\pi}{2} $. Find $ heta $. 📰 Discover The Secret Sentence Of Count That Boosts Your Content Like Magic 1176126 📰 Finally Revealed The True Power Of 200000 Yen To Usd 1254749 📰 5Why You Can Earn Big Money Overnightheres Your Step By Step Guide 7526993 📰 From Hero To Scandal Inside Yelena Belovas Rise And Spectacular Downfall 6378096 📰 Uber Eats Merchant 4108491 📰 From Wildfires To Breakthroughs Fox13Utahs Top 5 Must Know Stories Now 5377127 📰 Wells Fargo Investment Banking Deals 7394276 📰 Best Airline Loyalty Program 3562032 📰 Verizon Wireless Horseheads 9259335 📰 Trimui Brick Shocked Every Designer This One Trick Transformed My Workflow 2557181 📰 Danny Trejos Ultimate Tattoo Masterpiece Exposed The Meaning Behind Every Detail 9735471 📰 How To Avoid Microplastics 6265505 📰 The Inevitable Defeat Of 5203501 📰 You Didnt Just Like Itwhat Did It Really Do To You 2259194 📰 2 Minutes Of Genius The Qb Who Suddenly Redefined Greatness 7607824