$f:N\to A$ and $g:N\to B$ are two onto functions such that $f\left(n\right)=\left[n{\text{sec}}^2\theta \right]$  and $g\left(n\right)=\left[n{\text{cosec}}^2\theta \right],$ for some $\theta$ and $n\in N$ where N is set of natural numbers, if ${\text{sec}}^2\theta$ is irrational, (where[.] denotes the greatest integer function) then which of following is/are true 

Since ${\text{sec}}^2\theta >1\Rightarrow \left[(\text{n}+1){\text{sec}}^2\theta \right]>\left[{\text{nsec}}^2\theta \right]$ Hence, $f$ and $g$ are both one-one

Let  $\text{k}<{\text{nsec}}^2\theta <\text{k}+1$ and $\text{k}<{\text{mcosec}}^2\theta <\text{k}+1$

${\text{kcos}}^2\theta <\text{n}<(\text{k}+1){\text{cos}}^2\theta \text{and} {\text{ksin}}^2\theta <\text{m}<(\text{k}+1){\text{sin}}^2\theta$

$\text{Adding given} \text{k}< \text{integer} <\text{k}+1\Rightarrow \text{A}\cap \text{B}=\varphi$

Suppose $k\notin A\Rightarrow {\text{nsec}}^2\theta <\text{k} \text{and} (\text{k}+1)<(\text{n}+1){\text{sec}}^2\theta$

$\Rightarrow \text{n}<{\text{kcos}}^2\theta \text{and} \text{n}+1>(\text{k}+1){\text{cos}}^2\theta \Rightarrow \text{k}<(\text{k}-\text{n}){\text{cosec}}^2\theta ,\text{k}+1>(\text{k}-\text{n}){\text{cosec}}^2\theta$

 $\left[(k-n)\cos e{c}^2\theta \right]=k\in B$