# 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

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