Statement-I      $\cos e{c}^2\theta -{\cot }^2\theta$  is always equal to 1

Statement-II      $\cot \theta$ is always defined.

When  $\theta =0°$  or  $\pi$  

$opp=y=0$

$adj=x$

$hyp=r=x$

$\cos ec\theta =\frac{r}{y}=\frac{r}{0}$  which is undefined.

$\cot \theta =\frac{y}{x}=\frac{y}{0}$  which undefined.

Therefore  $\cos ec\theta$  and  $\cot \theta$  are not defined multiples  $\pi$

Domain of  $\cot \theta$  and  $\cos ec\theta =R-\left\{n\pi ,wheren\in Z\right\}$

$\therefore$   $\cos e{c}^2\theta -{\cot }^2\theta =1$  for all  $x\in R$  except  $\left\{x=n\pi ,n\in Z\right\}$