Question

answer the questions and I will mark it as brainlist answer​

Answer 1

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

$\rm :\longmapsto\:\boxed{ \tt{ \: tan\theta = \frac{1}{ \sqrt{7} } \: }}$

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\:\boxed{ \tt{ \: \frac{ {cosec}^{2}\theta - {sec}^{2}\theta}{ {cosec}^{2} \theta + {sec}^{2} \theta} \: }}$

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

Given that,

$\rm :\longmapsto\:tan\theta = \dfrac{1}{ \sqrt{7} }$

Now, Consider

$\rm :\longmapsto\:\dfrac{ {cosec}^{2}\theta - {sec}^{2}\theta}{{cosec}^{2}\theta + {sec}^{2}\theta}$

We know,

$\boxed{ \tt{ \: {cosec}^{2}\theta - {cot}^{2}\theta = 1}}$

and

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

So, on using these Identities,

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

$\rm \: = \:\dfrac{1 + {cot}^{2} \theta- 1 - {tan}^{2}\theta }{1 + {cot}^{2} \theta + 1 + {tan}^{2}\theta}$

$\rm \: = \:\dfrac{{cot}^{2} \theta - {tan}^{2}\theta }{2 + {cot}^{2} \theta + {tan}^{2}\theta}$

As,

$\rm :\longmapsto\:\boxed{ \tt{ \: tan\theta = \frac{1}{ \sqrt{7} } \: }}$

So,

$\boxed{ \tt{ \: cot\theta = \frac{1}{tan\theta}}} \: \: \rm \implies\:\boxed{ \tt{ \: cot\theta = \sqrt{7} \: }}$

So, on substituting these values, we get

$\rm \: = \:\dfrac{7 - \dfrac{1}{7} }{2 + \dfrac{1}{7} + 7}$

$\rm \: = \:\dfrac{\dfrac{49 - 1}{7} }{ \dfrac{14 + 1 + 49}{7}}$

$\rm \: = \:\dfrac{\dfrac{48}{7} }{ \dfrac{64}{7}}$

$\rm \: = \:\dfrac{48}{7} \times \dfrac{7}{64}$

$\rm \: = \:\dfrac{3}{4}$

Hence,

$\rm :\longmapsto\:\boxed{ \tt{ \: \: \frac{ {cosec}^{2}\theta - {sec}^{2}\theta}{ {cosec}^{2} \theta + {sec}^{2} \theta} \: = \: \frac{3}{4} \: \: }}$

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