If y-1-tan x-1-tan x- show that d y-d x-2-1-sin 2 x

By the quotinet rule, we have, dy/dx=(1+tan x),dy/dx(1 tan x).dy/dx.(1+tan x)/(1+tan x)^2 =(1 tan x)( sec^2 x) (1 tan x)(sec^2 x)/(1+tan x)^2 = 2sec^2 x/(1+tan ...

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Byju's Answer

Standard XII

Mathematics

Standard Formulae - 3

If y=1-tan x/...

Question

If $y = {\frac{1 - t a n x}{1 + t a n x}}, s h o w t h a t \frac{d y}{d x} = \frac{- 2}{(1 + s i n 2 x)}$

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Solution

By the quotinet rule, we have,

$\frac{d y}{d x} = \frac{(1 + t a n x), \frac{d y}{d x} (1 - t a n x) . \frac{d y}{d x} . (1 + t a n x)}{(1 + t a n x)^{2}}$

$= \frac{(1 - t a n x) (- s e c^{2} x) - (1 - t a n x) (s e c^{2} x)}{(1 + t a n x)^{2}}$ $= \frac{- 2 s e c^{2} x}{(1 + t a n x)^{2}} = \frac{- 2}{(c o s^{2} x) (1 + t a n^{2} x + 2 t a n x)}$

$= \frac{- 2}{(c o s^{2} x) {1 + \frac{s i n^{2} x}{c o s^{2} x} + \frac{2 s i n x}{c o s x}}} = \frac{- 2}{(1 + s i n 2 x)}$

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Similar questions

Q. If

y = l o g \sqrt{\frac{1 + tan x}{1 - tan x}}

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If y={1−tan x1+tan x},show that dydx=−2(1+sin 2x)

By the quotinet rule, we have, dydx=(1+tan x),dydx(1−tan x).dydx.(1+tan x)(1+tan x)2 =(1−tan x)(−sec2 x)−(1−tan x)(sec2 x)(1+tan x)2 =−2sec2 x(1+tan x)2=−2(cos2 x)(1+tan2 x+2 tan x) =−2(cos2 x){1+sin2 xcos2 x+2 sin xcos x}=−2(1+sin 2x)

If $y = {\frac{1 - t a n x}{1 + t a n x}}, s h o w t h a t \frac{d y}{d x} = \frac{- 2}{(1 + s i n 2 x)}$