Question

sin theta/cot theta+cosec theta + 2+sin theta/cot theta-cosec theta​

Answer 1

Appropriate Question :-

Prove that,

$\rm \: \dfrac{sin\theta }{cot\theta + cosec\theta } = 2 + \dfrac{sin\theta }{cot\theta - cosec\theta}$

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

Consider LHS

$\rm \: \dfrac{sin\theta }{cot\theta + cosec\theta }$

We know,

$\boxed{\sf{ \:cot\theta = \frac{cos\theta }{sin\theta } \: }}$

and

$\boxed{\sf{ \:cosec\theta = \frac{1}{sin\theta } \: }}$

So, on substituting these Identities, we get

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

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

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

We know,

$\boxed{\sf{ \: {sin}^{2}\theta + {cos}^{2}\theta = 1 \: }}$

So, using this identity, we get

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

$\rm \: = \: \dfrac{(1 - cos\theta)(1 + cos\theta ) }{1 + cos\theta }$

$\rm \: = \: 1 - cos\theta \:$

So,

$\rm \: \:\boxed{\sf{ \: \: \rm \: \dfrac{sin\theta }{cot\theta + cosec\theta } = 1 -cos\theta \: \: }} - - - (1)$

Now, Consider RHS

$\rm \: 2 + \dfrac{sin\theta }{cot\theta - cosec\theta}$

can be rewritten as

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

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

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

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

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

$\rm \: = \: 2 - (1 + cos\theta )$

$\rm \: = \: 2 - 1 - cos\theta$

$\rm \: = \: 1 - cos\theta$

So,

$\rm\implies \:\boxed{\sf{ \:\rm \: 2 + \dfrac{sin\theta }{cot\theta - cosec\theta} = 1 - cos\theta }} - - - (2) \:$

From equation (1) and (2), we concluded that

$\boxed{\sf{ \: \: \: \:\rm \: \dfrac{sin\theta }{cot\theta + cosec\theta } = 2 + \dfrac{sin\theta }{cot\theta - cosec\theta} \: \: \: \: \: }}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

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

Answer 2

Step-by-step explanation:

$\color{red}\tt \leadsto\dfrac{\sin \theta}{\cot \theta+\operatorname{cosec} \theta}=2+\dfrac{\sin \theta}{\cot \theta-\operatorname{cosec} \theta} \: \: \: is \: \: true$

$\tt \color{navy} \leadsto \: if \quad \dfrac{\sin \theta}{\cot \theta+\operatorname{cosec} \theta}-\dfrac{\sin \theta}{\cot \theta-\operatorname{cosec} \theta}=2 \: \: \: is \: \: \: \: true$

$\tt \color{blue} \leadsto \quad \: i.e. \: \: \: if \: \: \: \cfrac{\sin \theta(\cot \theta-\operatorname{cosec} \theta)-\sin \theta(\cot \theta+\operatorname{cosec} \theta)}{\cot ^{2} \theta-\operatorname{cosec}^{2} \theta}=2 \: \: \: is \: \: \: \: true$

$\tt \color{purple} \leadsto \quad \: i.e. \: \: if \: \: \: \dfrac{-2 \sin \theta \operatorname{cosec} \theta}{-1}=2 \: \: \: is \: \: \: true$

$\tt \color{cyan} \leadsto i.e. \: \: \: \: if \: \: \: \: 2=2 , \: \: \: \: which \: \: \: is \: \: \: true.$