The differential coefficient of the function | vedclass

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

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

The differential coefficient of the function $\log_e \left( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \right)$ with respect to $x$ is:

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