Answer
a. $(-\infty,\infty)$ except $x\ne k\pi+\frac{\pi}{2}$
b. $(-\infty,\infty)$ except $x\ne k\pi$
c. $(-\infty,\pi)\cup(\pi,\infty)$
d. $(-\infty,0)\cup(0,\infty)$
Work Step by Step
a. The function $f(x)=tan(x)$ has a domain of all real numbers except $x\ne k\pi+\frac{\pi}{2}$ and the function is continuous over all its domain values, that is $(-\infty,\infty)$ except $x\ne k\pi+\frac{\pi}{2}, k=0,\pm1,\pm2,...$
b. The function $g(x)=csc(x)$ has a domain of all real numbers except $x\ne k\pi$ and the function is continuous over all its domain values, that is $(-\infty,\infty)$ except $x\ne k\pi$
c. The function $h(x)=\frac{cos(x)}{x-\pi}$ has a domain of all real numbers except $x\ne \pi$ and the function is continuous over all its domain values, that is $(-\infty,\pi)\cup(\pi,\infty)$
d. The function $k(x)=\frac{sin(x)}{x}$ has a domain of all real numbers except $x\ne 0$ and the function is continuous over all its domain values, that is $(-\infty,0)\cup(0,\infty)$
