Question

sin theta upon 1+cos theta simplify it​

Answer 1

${ \huge{ \green{ \underline{ \red{ \mathbb{ AnSwEr{!!}}}}}}}$

THE CORRECT ANSWER IS OPTION B .

$\frac{\sin\theta}{1 \: + \: cos \ \theta} = \: \frac{\sin \ \theta(1 \: - \: \cos \ \theta)}{(1 \: + \: \cos \: \theta) \: (1 \: - \: \cos \ \theta)} \\ \\ \: \: \: \: \: \: = \frac{\sin \ \theta(1 \: - \: \cos \ \theta)}{1 \: - \: { \cos}^{2} \ \theta } \\ \\ \: \: \: \: \: \: \: = \frac{\sin \ \theta(1 \: - \: \cos \ \theta)}{ { \sin}^{2} \ \theta} \\ \\ = \: \frac{1 \: - \: cos \ \theta}{sin \ \theta}$

Answer 2

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

Given Trigonometric expression is

$\rm :\longmapsto\:\dfrac{sin\theta }{1 + cos\theta }$

So, on rationalizing the denominator, we get

$\rm \:  =  \: \dfrac{sin\theta }{1 + cos\theta } \times \dfrac{1 - cos\theta }{1 - cos\theta }$

We know,

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

So, using this, we get

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

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ {sin}^{2}x + {cos}^{2}x = 1}}}$

So, using this identity, we get

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

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

Hence,

$\\ \red{\rm\implies \:\boxed{\tt{ \rm \:\dfrac{sin\theta }{1 + cos\theta }   =  \: \dfrac{1 - cos\theta }{sin\theta } }}}$

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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