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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} $. 📰 Shocked These Tiny Miniature Stallions Steal Heartssee Why Theyre All The Trend 7980323 📰 These Preppy Wallpapers Are Taking Social Media By Storm Shop Now 1535432 📰 You Wont Believe What This Simple Trick Is Catching Online Overnight 7806113 📰 Albanian Girls 5997811 📰 The Iron Skillet Indianapolis 9591610 📰 Unlock The Secret To Stunning Hamster Drawings With These Simple Tips 8423906 📰 The Shampoo That Lasts Three Weeks But Cleans Deep Like Daily Treatmentsno More Stress 4292983 📰 Ramadan 2026 The Surprising Twist That Will Shock Select Communities 6630887 📰 Robert House 4074785 📰 Finn Mertens Adventure Time 3364146 📰 Neds Club Washington Dc 8504147 📰 Psd3 Psr News Today 5191381 📰 South Pine American Eatery 5047505 📰 Finally Found The Best Visio App Download For Beginners Pros Alike 7758334 📰 You Wont Believe What Happened When Yahoo Launched This Secret Feature 3483022