Word Count : 592

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We are asked to solve two problems: a differential equation and an estimation problem using the regression line. We'll handle them step by step. --- (a) Solve ((1 - \sin x \tan y) dx + (\cos x \sec^2 y) dy = 0). First, write it in the form: [ (1 - \sin x \tan y)dx + (\cos x \sec^2 y)dy = 0 \implies \frac{dy}{dx} = -\frac{1 - \sin x \tan y}{\cos x \sec^2 y}. ] Simplify (\sec^2 y = 1/\cos^2 y): [ \frac{dy}{dx} = -\frac{1 - \sin x \tan y}{\cos x / \cos^2 y} = -\frac{(1 - \sin x \tan y) \cos^2 y}{\cos x}. ] Now (\tan y = \sin y / \cos y), so (1 - \sin x \tan y = 1 - \sin x \frac{\sin y}{\cos y} = \frac{\cos y - \sin x \sin y}{\cos y}.) Multiply by (\cos^2 y): [ \frac{dy}{dx} = - \frac{(\cos y _____ _____ _____ _______ __________ ____ __________ _____.
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