Question

please answer this equation​

Answer 1

Answer:

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Answer 2

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

Given that

$\rm :\longmapsto\:cos\theta = \dfrac{ \sqrt{4 - {x}^{2} } }{2}$

We know that,

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

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

$\rm :\longmapsto\: {sin}^{2}\theta \: = 1 - \dfrac{4 - {x}^{2} }{4}$

$\rm :\longmapsto\: {sin}^{2}\theta \: = \dfrac{4 - (4 - {x}^{2}) }{4}$

$\rm :\longmapsto\: {sin}^{2}\theta \: = \dfrac{4 - 4 + {x}^{2}}{4}$

$\rm :\longmapsto\: {sin}^{2}\theta \: = \dfrac{{x}^{2}}{4}$

$\bf\implies \:sin\theta = \dfrac{x}{2}$

Now, we know that,

$\purple{\rm :\longmapsto\:sec\theta = \dfrac{1}{cos\theta } }$

$\bf\implies \:sec\theta = \dfrac{4}{ \sqrt{4 - {x}^{2} } }$

Also,

$\purple{\rm :\longmapsto\:tan\theta = \dfrac{sin\theta }{cos\theta } = \dfrac{\dfrac{x}{2} }{\dfrac{ \sqrt{4 - {x}^{2} } }{4} } = \dfrac{2x}{ \sqrt{4 - {x}^{2} } } }$

Also,

$\purple{\rm :\longmapsto\:cot\theta = \dfrac{1}{tan\theta } = \dfrac{ \sqrt{4 - {x}^{2} } }{2x}}$

Also,

$\purple{\rm :\longmapsto\:cosec\theta = \dfrac{1}{sin\theta } = \dfrac{2}{x} }$

Consider (i)

$\rm :\longmapsto\:sec\theta \: tan\theta$

On substituting the values, we get

$\rm \: = \: \:\dfrac{4}{ \sqrt{4 - {x}^{2} } } \times \dfrac{2x}{ \sqrt{4 - {x}^{2} } }$

$\rm \: = \: \:\dfrac{8x}{4 - {x}^{2} }$

Consider (ii)

$\rm :\longmapsto\:cosec\theta \: cot\theta$

$\rm \: = \: \:\dfrac{2}{x} \times \dfrac{ \sqrt{4 - {2}^{2} } }{2x}$

$\rm \: = \: \: \dfrac{ \sqrt{4 - {2}^{2} } }{ {x}^{2} }$

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