NIMCET 2022 — Mathematics PYQ

Solution 
1. The given equation is $\text{cosec }\theta -\text{cot }\theta =2$. 
2. The trigonometric identity ${\text{cosec}}^2\theta -{\text{cot}}^2\theta =1$ is used. 
3. The identity can be factored as $(\text{cosec }\theta -\text{cot }\theta )(\text{cosec }\theta +\text{cot }\theta )=1$. 
4. Substituting the given value into the factored identity, $(2)(\text{cosec }\theta +\text{cot }\theta )=1$ is obtained. 
5. Solving for $\text{cosec }\theta +\text{cot }\theta$, the equation $\text{cosec }\theta +\text{cot }\theta =\frac{1}{2}$ is found. 
6. Now, a system of two linear equations is formed: 

$\text{cosec }\theta -\text{cot }\theta =2\text{(Equation 1)}$ $\text{cosec }\theta +\text{cot }\theta =\frac{1}{2}\text{(Equation 2)}$ 

1. Adding Equation $1$ and Equation $2$ eliminates $\text{cot }\theta$: 

$(\text{cosec }\theta -\text{cot }\theta )+(\text{cosec }\theta +\text{cot }\theta )=2+\frac{1}{2}$ $2\text{ cosec }\theta =\frac{4}{2}+\frac{1}{2}$ $2\text{ cosec }\theta =\frac{5}{2}$ 

1. Solving for $\text{cosec }\theta$, the value $\text{cosec }\theta =\frac{5}{4}$ is determined. 

Final Answer 
The value of $\text{cosec }\theta$ is $\frac{5}{4}$.