If `y=logsqrt((1+tanx)/(1-tanx)),"P r o v et h a t"(dy)/(dx)=sec2x`

Explanation

`y=log sqrt{frac{1+tan x}{1-tan x}} `

or, `y=frac{1}{2} log frac{(1+tan x)}{(1-tan x)} `

or, `y=frac{1}{2}(log (1+tan x)-log (1-tan x))`

`frac{d y}{d x}=frac{1}{2}(frac{sec ^{2} x}{1+tan x}-frac{-sec ^{2} x}{1-tan x})`

or, `frac{d y}{d x}=frac{1}{2}(frac{sec ^{2} x}{1+tan x}+frac{sec ^{2} x}{1-tan x})`

or, `frac{d y}{d x}=frac{1}{2}(frac{2 sec ^{2} x}{1-tan ^{2} x})`

or, `frac{{dy}}{{dx}}=(frac{1+tan ^{2} {x}}{1-tan ^{2} {x}})`

or, `frac{d y}{d x}=frac{1}{cos 2 x}`

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