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Simplify: 1cosec⁡θ−cot⁡θ\frac{1}{\cosec \theta - \cot \theta}cosecθ−cotθ1​​​
Question

Simplify: 1cosec⁡θ−cot⁡θ\frac{1}{\cosec \theta - \cot \theta}​​

A.

cosec θ + cot θ

B.

sin θ + cos θ

C.

cosec2θ + cos2 θ

D.

cosec θ - cot θ

Correct option is A

Given: 
​1cosec⁡θ−cot⁡θ\frac{1}{\cosec\theta -\cot \theta} 
Formula Used: 
​1+cot⁡2θ=cosec⁡2θ1+\cot^2 \theta = \cosec^2 \theta​​
Solution:
​1(cosec⁡θ−cot⁡θ)×(cosec⁡θ+cot⁡θ)(cosec⁡θ+cot⁡θ) This yields: =(cosec⁡θ+cot⁡θ)(cosec⁡θ)2−(cot⁡θ)2 =(cosec⁡θ+cot⁡θ)1 =cosec⁡θ+cot⁡θ\frac{1}{\left(\cosec \theta - \cot \theta\right)} \times \frac{\left(\cosec \theta + \cot \theta\right)}{\left(\cosec \theta + \cot \theta\right)} \\ \ \\\text{This yields:} \\ \ \\= \frac{\left(\cosec \theta + \cot \theta\right)}{\left(\cosec \theta\right)^2 - \left(\cot \theta\right)^2} \\ \ \\= \frac{\left(\cosec \theta + \cot \theta\right)}{1} \\ \ \\= \cosec \theta + \cot \theta​​

​

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