Question

cos0-sin0+1/cos0+sin0+1=cot0+cosec0​

Answer 1

The answer is proved above in the image

Answer 2

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

The Correct Statement is

$\sf \: Prove \: that \: \dfrac{cos \theta \: - sin\theta \: + 1 }{cos\theta \: + sin\theta \: - 1} = cosec\theta \: + cot\theta \:$

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

Identities Used :-

$1. \: \: \: \boxed{ \bf{cot\theta \: = \dfrac{cos\theta \:}{sin\theta \:} }}$

$2. \: \: \: \boxed{ \bf{cosec\theta \: = \dfrac{1}{sin\theta \:} }}$

$3. \: \: \: \boxed{ \bf{ {cosec}^{2}\theta \: - {cot}^{2}\theta \: = 1}}$

$4. \: \: \: \boxed{ \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}}$

Lets do the problem now!!

$\bf \: Consider \: LHS -$

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

Divide numerator and denominator by sinθ, w get

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

$\sf \: = \: \dfrac{cot\theta \: - 1 + cosec\theta \:}{cot\theta \: + 1 - cosec\theta \:}$

$\sf \: = \: \dfrac{(cosec\theta \: + cot\theta \:) - 1}{cot\theta \: + 1 - cosec\theta \:}$

$\sf \: = \: \dfrac{(cosec\theta \: + cot\theta \:) - ( {cosec}^{2}\theta \: - {cot}^{2}\theta \:) }{cot\theta \: - cosec\theta \: + 1}$

$\sf \: = \: \dfrac{(cosec\theta \: + cot\theta \:) - (cosec\theta \: + cot\theta \:)(cosec\theta \: - cot\theta \:)}{cot\theta \: - cosec\theta \: + 1}$

Take out cosecθ + cotθ common from numerator, we get

$\sf \: = \: \dfrac{(cosec\theta \: + cot\theta \:) \: \cancel{(1 - cosec\theta \: + cot\theta \:)}}{ \cancel{cot\theta \: - cosec\theta \: + 1}}$

$\sf \: = \: cosec\theta \: + cot\theta \:$

$\bf \: = \: RHS$

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

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