Question

$\footnotesize\sf{ Prove \: the \: following \: identities }$

$\: \: \: \: \: \: \: \: \: \: \: \:$
$\sf{ \frac{1}{cosec \: A\: - \: cot \: A} - \frac{1}{sin \:A } = \frac{1}{sin \:A } - \frac{1}{cosec \: A \: + \: cot \: A} }$
$\: \: \: \: \: \: \: \: \: \: \:$
$\footnotesize\sf{ Don't \: Spam}$


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Answer 1

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

Consider LHS

$\rm :\longmapsto\:\dfrac{1}{cosec \: A\: - \: cot \: A} - \dfrac{1}{sin \:A }$

On rationalizing the denominator, we get

$\rm \: = \: \dfrac{1}{cosecA - cotA} \times \dfrac{cosecA + cotA}{cosecA + cotA} - cosecA$

We know,

$\boxed{ \tt{ \: (x + y)(x - y) = {x}^{2} - {y}^{2} \: }}$

So, using this, we get

$\rm \: = \: \dfrac{cosecA + cotA }{{cosec}^{2} A - {cot}^{2} A} - cosecA$

We know,

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

So, using this, we get

$\rm \: = \: \dfrac{cosecA + cotA}{1} - cotA$

$\rm \: = \: cosecA + cotA - cosecA$

$\rm \: = \: cotA$

Thus,

$\red{\rm \implies\:\boxed{ \tt{ \: \dfrac{1}{cose A - cot A} - \dfrac{1}{sinA } = cotA \: }} - - (1)}$

Now, Consider RHS

$\rm :\longmapsto\:\dfrac{1}{sinA} - \dfrac{1}{cosecA + cotA}$

On rationalizing the denominator, we get

$\rm \: = \: cosecA - \dfrac{1}{cosecA + cotA} \times \dfrac{cosecA - cotA}{cosecA - cotA}$

$\rm \: = \: cosecA - \dfrac{cosecA - cotA}{ {cosec}^{2}A - {cot}^{2}A}$

$\rm \: = \: cosecA - \dfrac{cosecA - cotA}{1}$

$\rm \: = \: cosecA - (cosecA - cotA)$

$\rm \: = \: cosecA - cosecA + cotA$

$\rm \: = \: cotA$

Thus,

$\red{\rm \implies\:\boxed{ \tt{ \: \dfrac{1}{sinA} - \dfrac{1}{cosecA + cotA} = cotA}} - - (2)}$

Thus, from equation (1) and (2), we concluded that

$\boxed{ \tt{ \: \frac{1}{cosecA + cotA} - \frac{1}{sinA} = \frac{1}{sinA} - \frac{1}{cosecA - cotA} \: }}$

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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

Answer:

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