Statement (A):  $\cos e{c}^2\theta -{\cot }^2\theta$ is always equal to 1

            Statement (B):  $\cot \theta$ 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   $\cos ec\theta =R-\{n\pi ,wheren\in Z\}$

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