If y = frac{sin x}{1 + frac{cos x}{1 + frac{s | vedclass

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Question

(A) Given the infinite continued fraction: $y = \frac{\sin x}{1 + \frac{\cos x}{1 + y}}$.Multiplying both sides by $(1 + y + \cos x)$,we get: $y(1 + y

Vedclass · Accepted Answer

$\frac{y \sin x + (1 + y) \cos x}{1 + 2y + \cos x - \sin x}$

Vedclass · Answer

(A) Given the infinite continued fraction: $y = \frac{\sin x}{1 + \frac{\cos x}{1 + y}}$.Multiplying both sides by $(1 + y + \cos x)$,we get: $y(1 + y + \cos x) = \sin x(1 + y)$.$y + y^2 + y \cos x = \sin x + y \sin x$.Differentiating both sides with respect to $x$ using the product rule:$\frac{dy}{dx} + 2y \frac{dy}{dx} + (\frac{dy}{dx} \cos x - y \sin x) = \cos x(1 + y) + \sin x \frac{dy}{dx}$.Rearranging the terms to isolate $\frac{dy}{dx}$:$\frac{dy}{dx}(1 + 2y + \cos x - \sin x) = \cos x + y \cos x + y \sin x$.$\frac{dy}{dx} = \frac{y \sin x + (1 + y) \cos x}{1 + 2y + \cos x - \sin x}$.

If $y = \frac{\sin x}{1 + \frac{\cos x}{1 + \frac{\sin x}{1 + \dots}}}$,then $\frac{dy}{dx}$ is given by

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