Question

Solve the equation : cos theta /1-sin theta + cos theta / 1+ sin theta please don't spam don't copy quality answers needed​

Answer 1

Step-by-step explanation:

its 2 sec theta . the answer is provided. hope you understand it

Answer 2

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

Given Trigonometric function is

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

Taking common, we have

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

On taking LCM, we get

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

We know,

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

So, using this identity, we get

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

We know,

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

So, using this identity, we get

$\rm \:  =  \:cos\theta \times \bigg[\dfrac{2}{ {cos}^{2}\theta} \bigg]$

$\rm \:  =  \:\dfrac{2}{cos\theta }$

$\rm \:  =  \:2 \: sec\theta$

Hence,

$\green{\rm \implies\:\boxed{ \tt{ \: \dfrac{cos\theta }{1 - sin\theta } + \dfrac{cos\theta }{1 + sin\theta } = 2 \: sec\theta }}}$

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