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Answer 1

Given Question :-

$\sf \: \sqrt{\dfrac{1 - cos\theta }{1 + cos\theta } } \times \sqrt{\dfrac{cosec\theta - cot\theta }{cosec\theta + cot\theta } } = \dfrac{r - 1}{r + 1}$

then

$\: \: \: \: (1) \: tan \theta \: = \: \sqrt{ {r}^{2} - 1}$

$\: \: \: \: (2) \: cos \theta \: = \: r$

$\: \: \: \: (3) \:sin \theta \: + cos \theta \: = \: \dfrac{ \sqrt{1 + {r}^{2} } }{r}$

$\: \: \: \: (4) \: cot \theta \: = \: \sqrt{1 - {r}^{2}}$

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

Given Trigonometric equation

$\rm :\longmapsto\:\sf \:\sqrt{\dfrac{1 - cos\theta }{1 + cos\theta } } \times \sqrt{\dfrac{cosec\theta - cot\theta }{cosec\theta + cot\theta } } = \dfrac{r - 1}{r + 1}$

$\rm :\longmapsto\:\sf \:\sqrt{\dfrac{1 - cos\theta }{1 + cos\theta } } \times \sqrt{\dfrac{\dfrac{1}{sin\theta } - \dfrac{cos\theta }{sin\theta } }{\dfrac{1}{sin\theta } + \dfrac{cos\theta }{sin\theta } } } = \dfrac{r - 1}{r + 1}$

$\rm :\longmapsto\:\sf \:\sqrt{\dfrac{1 - cos\theta }{1 + cos\theta } } \times \sqrt{\dfrac{\dfrac{1 - cos\theta }{sin\theta } }{\dfrac{1 + cos\theta }{sin\theta }} } = \dfrac{r - 1}{r + 1}$

$\rm :\longmapsto\:\sf \:\sqrt{\dfrac{1 - cos\theta }{1 + cos\theta } } \times\sqrt{\dfrac{1 - cos\theta }{1 + cos\theta } } = \dfrac{r - 1}{r + 1}$

$\rm :\longmapsto\:\sf \:\dfrac{1 - cos\theta }{1 + cos\theta} = \dfrac{r - 1}{r + 1}$

can be rewritten as

$\rm :\longmapsto\:\sf \:\dfrac{1 + cos\theta }{1 - cos\theta} = \dfrac{r + 1}{r - 1}$

On applying Componendo and Dividendo, we get

$\rm :\longmapsto\:\sf \:\dfrac{1 + cos\theta + 1 - cos\theta }{1 + cos\theta - 1 + cos\theta} = \dfrac{r + 1 + r - 1}{r + 1 - r + 1}$

$\rm :\longmapsto\:\sf \:\dfrac{2 }{2cos\theta} = \dfrac{2r}{2}$

$\rm :\longmapsto\:\sf \:\dfrac{1}{cos\theta} = r$

$\rm \implies\:r = sec\theta$

$\rm \implies\: {r}^{2} = sec^{2} \theta$

$\rm \implies\: {r}^{2} = 1 + tan^{2} \theta$

$\rm \implies\: {r}^{2} - 1 = tan^{2} \theta$

$\bf\implies \:tan\theta = \sqrt{ {r}^{2} - 1 }$

So, Option (1) is correct.

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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