Question

If sin θ = 2x and cos θ = 2/x , 0⁰< θ< 90⁰, then the value of 2(x² + 1/x² ) is:
a)1
b) 2
c) 1/2
d) 1/4
e)Option 1

Answer 1

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

Given that,

$\rm :\longmapsto\:sin \theta = 2x$

and

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

We know,

$\rm :\longmapsto\: {sin }^{2}\theta + {cos}^{2}\theta = 1$

On substituting the values, we get

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

$\rm :\longmapsto\: {4x}^{2} + \dfrac{4}{ {x}^{2} } = 1$

$\rm :\longmapsto\:4\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) = 1$

$\bf\implies \:2\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} }\bigg) = \dfrac{1}{2}$

$\green{\rm\implies \:\boxed{\tt{ \: \: \: Option \: (c) \: is \: correct \: \: \: }}}$

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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

Answer 2

• options c is correct.

$\huge\mathfrak\red{Thanks}$