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📰 Solution: Let $ y = \sin(2x) $. The equation becomes $ y^2 + 3y + 2 = 0 $, which factors as $ (y + 1)(y + 2) = 0 $. Thus, $ y = -1 $ or $ y = -2 $. Since $ \sin(2x) $ ranges between $-1$ and $1$, $ y = -2 $ is invalid. For $ y = -1 $, $ \sin(2x) = -1 $ has infinitely many solutions (e.g., $ 2x = rac{3\pi}{2} + 2\pi k $, $ k \in \mathbb{Z} $). However, if restricted to a specific interval (not stated here), the count would depend on the domain. Assuming $ x \in \mathbb{R} $, there are infinitely many solutions. But if the question implies a general count, the answer is oxed{ ext{infinitely many}}. 📰 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} $. 📰 Sakura Swim Club 📰 Nfl Thursday Night Football Schedule 8459897 📰 Life Policy Quote 📰 Government Announces How To Be Nonchalant And It Sparks Debate 📰 Fortnight Free To Play 📰 History Museum St Petersburg 6084385 📰 Skip Microsoft Account Windows 11 7934747 📰 Juegos De Steam Gratuitos 📰 3 This Daily Difference Will Blow Your Mind The Secret That Worsens Or Improves Your Day Before You Know It 8773687 📰 No Signal Discover The Shocking Reasons Your Monitor Is Completely Black 5374636 📰 Viral Moment Arsmf Stock Price And It S Alarming 📰 The Ultimate Collection Ten Pentacles That Change Everything 3376066 📰 Discover The Hidden Secrets Keeping That Tree Stump A Treasure Only You Would Know 4921050 📰 Autogynephilia 📰 Small But Perfect Discovering The Best Piccole Cucine For Urban Spaces 9745202