Question

refer to the attachment ​

Answer 1

Answer:

Option (3) 2/sinθ is the correct option

Step-by-step explanation:

Given :

$\mathrm{\dfrac{1}{cosec\theta - cot\theta} + \dfrac{1}{cosec\theta + cot\theta} }$

To find :

The correct answer from the following options :

$(1) \ \mathrm{\dfrac{2}{cos\theta}}$

$(2) \ \mathrm{\dfrac{2}{cot\theta}}$

$(3) \ \mathrm{\dfrac{2}{sin\theta}}$

$(4) \ \mathrm{\dfrac{2}{cosec\theta}}$

Solution :

The given expression is :

$\longmapsto \mathrm{\dfrac{1}{cosec\theta - cot\theta} + \dfrac{1}{cosec\theta + cot\theta} }$

Taking LCM :

$\implies \mathrm{\dfrac{cosec\theta + cot\theta+cosec\theta-cot\theta}{(cosec\theta-cot\theta)(cosec\theta + cot\theta)} }$

$\text{We know that, } \boxed{\mathrm{(a+b)(a-b) = a^2-b^2}}$

Here, a = cosecθ ; b = cotθ

So, equation changes as,

$\implies \mathrm{\dfrac{2cosec\theta}{cosec^2\theta - cot^2\theta} }$

We know that,

$\boxed{\mathrm{cosec^2\theta - cot^2\theta = 1} }$

So,

$\implies \dfrac{2cosec\theta}{1}$

we know that,

$\boxed{\mathrm{cosec\theta = \dfrac{1}{sin\theta} }}$

Thus,

$\implies \mathrm{\dfrac{2}{sin\theta}}$

$\therefore \underline{\boxed{\mathrm{\dfrac{1}{cosec\theta - cot\theta} + \dfrac{1}{cosec\theta + cot\theta} } = \dfrac{2}{sin\theta}}}$

Thus, Option (3) is correct.

Hope it helps!!