Question

Prove that:

1 ÷ (SecA - TanA) = (SinA - CosA + 1) ÷
(SinA + CosA-1)

$1 \div {\sec(a) - \tan(a) } = {\sin(a) - \cos(a) + 1} \div {\sin(a) + \cos(a) - 1}$

LHS should be converted to RHS​

Answer 1

Identities Used :-

$\boxed{ \tt{ \: tanx = \frac{sinx}{cosx} \: }}$

$\boxed{ \tt{ \: secx = \frac{1}{cosx} \: }}$

$\boxed{ \tt{ \: {sec}^{2}x - {tan}^{2}x = 1 \: }}$

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

Consider RHS

$\rm :\longmapsto\:\dfrac{sin \alpha - cos\alpha + 1 }{sin\alpha + cos\alpha - 1}$

can be rewritten as

$\rm \:  =  \: \dfrac{\dfrac{sin\alpha - cos\alpha + 1}{cos\alpha} }{\dfrac{sin\alpha + cos\alpha - 1}{cos\alpha} }$

$\rm \:  =  \: \dfrac{\dfrac{sin\alpha}{cos\alpha} - \dfrac{cos\alpha}{cos\alpha} + \dfrac{1}{cos\alpha} }{\dfrac{sin\alpha}{cos\alpha} + \dfrac{cos\alpha}{cos\alpha} - \dfrac{1}{cos\alpha} }$

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

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

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

$\rm \:  =  \: \dfrac{tan\alpha + sec\alpha - ( sec\alpha + tan\alpha)(sec\alpha - tan\alpha)}{tan\alpha + 1 - sec\alpha}$

$\rm \:  =  \: \dfrac{(tan\alpha + sec\alpha)[1 - (sec\alpha - tan\alpha)]}{tan\alpha + 1 - sec\alpha}$

$\rm \:  =  \: \dfrac{(tan\alpha + sec\alpha)[1 - sec\alpha + tan\alpha]}{tan\alpha + 1 - sec\alpha}$

$\rm \:  =  \: sec\alpha + tan\alpha$

can be rewritten as

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

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

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

Hence,

$\rm :\longmapsto\:\boxed{ \tt{ \: \dfrac{sin \alpha - cos\alpha + 1 }{sin\alpha + cos\alpha - 1} = \frac{1}{sec\alpha - tan\alpha} \: }}$

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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