Question

Prove that: (1 + sin A - cos A)/(1 + sin A + cos A) = sqrt((1 - cos A)/(1 + cos A))​

Answer 1

Identities Used :-

$\boxed{ \red{ \bf \: \dfrac{sinx}{cosx} = tanx}}$

$\boxed{ \red{ \bf \: \dfrac{1}{cosx} = secx}}$

$\boxed{ \red{ \bf \: \dfrac{1}{sinx} = cosecx}}$

$\boxed{ \red{ \bf \: {sin}^{2} x + {cos}^{2} x = 1}}$

$\boxed{ \red{ \bf \: {cosec}^{2} x - {cot}^{2} x = 1}}$

$\large\underline{\sf{Solution-}}$

Consider,

$\rm :\longmapsto\:\dfrac{1 + sinA - cosA}{1 + sinA + cosA}$

On dividing numerator and denominator by sinA, we get

$\rm :\longmapsto\: \: = \: \:\dfrac{\dfrac{1}{sinA} + \dfrac{sinA}{sinA} - \dfrac{cosA}{sinA} }{\dfrac{1}{sinA} + \dfrac{sinA}{sinA} + \dfrac{cosA}{sinA} }$

$\rm :\longmapsto\: \: = \: \:\dfrac{cosecA + 1 - cotA}{cosecA +1 + cotA}$

$\rm :\longmapsto\: \: = \: \:\dfrac{cosecA - cotA + ( {cosec}^{2}A - {cot}^{2}A)}{cosecA + cotA + 1}$

$\rm :\longmapsto\: \: = \: \:\dfrac{(cosecA - cotA) + (cosecA + cotA)({cosec}A - {cot}A)}{cosecA + cotA + 1}$

$\rm :\longmapsto\: \: = \: \:\dfrac{(cosecA - cotA)(cosecA + cotA + 1)}{cosecA + cotA + 1}$

$\rm :\longmapsto\: \: = \: \:cosecA - cotA$

$\rm :\longmapsto\: \: = \: \: \sqrt{ {(cosecA - cotA)}^{2} }$

$\rm :\longmapsto\: \: = \: \: \sqrt{ {\bigg(\dfrac{1}{sinA} - \dfrac{cosA}{sinA} \bigg) }^{2} }$

$\rm :\longmapsto\: \: = \: \: \sqrt{ {\bigg(\dfrac{1 - cosA}{sinA}\bigg) }^{2} }$

$\rm :\longmapsto\: \: = \: \: \sqrt{\dfrac{ {(1 - cosA)}^{2} }{ {sin}^{2}A } }$

$\rm :\longmapsto\: \: = \: \: \sqrt{\dfrac{ {(1 - cosA)}^{2} }{1 - {cos}^{2}A } }$

$\rm :\longmapsto\: \: = \: \: \sqrt{\dfrac{ {(1 - cosA)}^{2} }{(1 - {cos}A)(1 + cosA) }}$

$\rm :\longmapsto\: \: = \: \: \sqrt{\dfrac{ (1 - cosA) }{(1 + cosA) }}$

$\rm :\longmapsto\: \: = \: \: \sqrt{\dfrac{ 1 - cosA}{1 + cosA}}$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1