frac{d}{dx} left( log sqrt{frac{1+sin x}{1-si | vedclass

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(D) Let $y = \log \sqrt{\frac{1+\sin x}{1-\sin x}}$. Using logarithmic properties,$y = \frac{1}{2} \log \left( \frac{1+\sin x}{1-\sin x} \right) = \fr

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(D) Let $y = \log \sqrt{\frac{1+\sin x}{1-\sin x}}$. Using logarithmic properties,$y = \frac{1}{2} \log \left( \frac{1+\sin x}{1-\sin x} \right) = \frac{1}{2} [\log(1+\sin x) - \log(1-\sin x)]$. Differentiating with respect to $x$: $\frac{dy}{dx} = \frac{1}{2} \left[ \frac{1}{1+\sin x} \cdot \cos x - \frac{1}{1-\sin x} \cdot (-\cos x) \right]$ $\frac{dy}{dx} = \frac{1}{2} \left[ \frac{\cos x}{1+\sin x} + \frac{\cos x}{1-\sin x} \right]$ $\frac{dy}{dx} = \frac{1}{2} \cos x \left[ \frac{1-\sin x + 1+\sin x}{(1+\sin x)(1-\sin x)} \right]$ $\frac{dy}{dx} = \frac{1}{2} \cos x \left[ \frac{2}{1-\sin^2 x} \right] = \frac{\cos x}{\cos^2 x} = \sec x$.

$\frac{d}{dx} \left( \log \sqrt{\frac{1+\sin x}{1-\sin x}} \right) = $

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