Question

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Answer 1

Given Question

$\sf \: If \: 3x = cosec \theta \: and \: \dfrac{3}{x} = cot\theta, \: then \: the \: value \: of \: 3\bigg( {x}^{2} - \frac{1}{ {x}^{2} }\bigg) \: is$

$\: \: \: \: \: \: \: \: (a) \: \: \dfrac{1}{2}$

$\: \: \: \: \: \: \: \: (b) \: \: \dfrac{1}{3}$

$\: \: \: \: \: \: \: \: (c) \: \: \dfrac{3}{4}$

$\: \: \: \: \: \: \: \: (d) \: \: \dfrac{2}{3}$

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

Given that,

$\rm :\longmapsto\:3x = cosec\theta$

and

$\rm :\longmapsto\:\dfrac{3}{x} = cot\theta$

We know that,

$\rm :\longmapsto\: {cosec}^{2}\theta - {cot}^{2}\theta = 1$

$\rm :\longmapsto\: {(3x)}^{2} - {\bigg[\dfrac{3}{x} \bigg]}^{2} = 1$

$\rm :\longmapsto\: {9x}^{2} - \dfrac{9}{ {x}^{2} } = 1$

$\rm :\longmapsto\:3\bigg(3 {x}^{2} - \dfrac{3}{ {x}^{2} }\bigg) = 1$

$\bf\implies \:3 {x}^{2} - \dfrac{3}{ {x}^{2} }= \dfrac{1}{3}$

So, Option (b) is correct

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Explore More

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