Find the derivative of tan^-1(cos x/1+sin x) with respect to sec^-1(x).

\(text{Let,} u = tan^{- 1} left( frac{cos x}{1 + sin x} right)\) \(Let u = tan^{- 1} left[ tanleft( frac{pi}{4} – frac{x}{2} right) right]\) \(Rightarrow u = frac{pi}{4} – frac{x}{2}\)

Differentiating it with respect to x

\(frac{du}{dx} = 0 – left( frac{1}{2} right)\) \(frac{du}{dx} = – frac{1}{2} . . . left( i right)\) \(text { Let, v } = se c^{- 1} x\)

Differentiating it with respect to x

\(frac{dv}{dx} = frac{1}{xsqrt{x^2 – 1}} . . . left( ii right)\) \(text { Dividing equation } left( i right) text { by}left( ii right),\) \(frac{frac{du}{dx}}{frac{dv}{dx}} = – frac{1}{2} times frac{xsqrt{x^2 – 1}}{1}\) \(frac{du}{dv} = frac{- xsqrt{x^2 – 1}}{2}\)

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