Question

simplify the following expression cosec theta ( 1+ cos theta ) ( cosec theta - cot theta )​

Answer 1

Answer:

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Answer 2

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

Given expression is

$\rm :\longmapsto\:cosec\theta (1 + cos\theta )(cosec\theta - cot\theta )$

We know,

$\boxed{ \bf{ \: cosecx = \frac{1}{sinx}}}$

and

$\boxed{ \bf{ \: cotx = \frac{cosx}{sinx}}}$

So, using these Identities, we get

$\rm \: = \:\dfrac{1}{sin\theta}(1 + cos\theta )\bigg[\dfrac{1}{sin\theta } - \dfrac{cos\theta }{sin\theta } \bigg]$

$\rm \: = \:\dfrac{1}{sin\theta}(1 + cos\theta )\bigg[\dfrac{1 - cos\theta }{sin\theta } \bigg]$

$\rm \: = \:\dfrac{1}{sin\theta}\bigg[\dfrac{(1 + cos\theta )(1 - cos\theta)}{sin\theta } \bigg]$

We know,

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

$\rm \: = \:\dfrac{1 - {cos}^{2}\theta }{ {sin}^{2}\theta }$

We know,

$\boxed{ \bf{ \: {sin}^{2}x + {cos}^{2}x = 1}}$

So, using this,

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

$\rm \: = \:1$

Hence,

$\: \boxed{ \: \: \: \bf{ \: cosec\theta (1 + cos\theta )(cosec\theta - cot\theta ) = 1 \: \: \: }}$

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