Question

sec alpha -tan alpha/sec alpha +tan alpha =1-2sec alpha tan alpha +2tan²alpha​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

Prove that,

$\rm :\longmapsto\:\dfrac{sec \alpha - tan\alpha }{sec\alpha + tan\alpha} = 1 - 2sec\alpha \: tan\alpha \: + \: {2tan}^{2}\alpha$

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

Consider LHS

$\rm :\longmapsto\:\dfrac{sec\alpha - tan\alpha }{sec\alpha + tan\alpha }$

On rationalizing the denominator, we get

$\rm \:  =  \:  \: \dfrac{sec\alpha - tan\alpha }{sec\alpha + tan\alpha } \times \dfrac{sec\alpha - tan\alpha }{sec\alpha - tan\alpha }$

We know,

$\green{\boxed{ \bf{ (x + y)(x - y) = {x}^{2} - {y}^{2}}}}$

So, using this identity, we get

$\rm \:  =  \:  \: \dfrac{ {(sec\alpha - tan\alpha )}^{2} }{ {sec}^{2}\alpha - {tan}^{2}\alpha }$

We know,

$\green{\boxed{ \bf{ {sec}^{2}x - {tan}^{2}x = 1}}}$

and

$\boxed{ \rm{ {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy}}$

Using these Identities, we get

$\rm \:  =  \:  \: \dfrac{ {sec}^{2}\alpha + {tan}^{2} \alpha - 2sec\alpha tan\alpha }{1}$

$\rm \:  =  \:  \: {sec}^{2}\alpha + {tan}^{2}\alpha - 2sec\alpha tan\alpha$

$\rm \:  =  \:  \: 1 + {tan}^{2}\alpha + {tan}^{2}\alpha - 2sec\alpha tan\alpha$

$\rm \:  =  \:  \: 1 + 2{tan}^{2}\alpha - 2sec\alpha tan\alpha$

$\rm \:  =  \:  \: 1 - 2sec\alpha tan\alpha + 2 {tan}^{2} \alpha$

Hence,

$\boxed{ \rm{ \:\dfrac{sec \alpha - tan\alpha }{sec\alpha + tan\alpha} = 1 - 2sec\alpha \: tan\alpha \: + \: {2tan}^{2}\alpha}}$

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