Question

show that sin theta - cos theta + 1 by sin theta + cos theta - 1 = sin theta + tan theta ​

Answer 1

Appropriate Question

$\rm :\longmapsto\:\dfrac{sin\theta - cos\theta + 1}{sin\theta + cos\theta - 1} = sec\theta + tan\theta$

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

Consider LHS

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

can be rewritten as

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

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

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

can be rewritten as

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

We know,

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

So, replacing 1 of numerator by this identity, we get

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

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

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

$\rm \: = \: sec\theta + tan\theta$

Hence,

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

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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