$$\eqalign{
& a)\,\,y = {3 \over {2\cos x}} \cr
& DKXD:\,\,\cos x \ne 0 \Leftrightarrow x \ne {\pi \over 2} + k\pi \,\,\left( {k \in Z} \right) \cr
& \Rightarrow D = R\backslash \left\{ {{\pi \over 2} + k\pi ;\,\,k \in Z} \right\} \cr
& b)\,\,y = cot\left( {2x - {\pi \over 4}} \right) \cr
& DKXD:\,\,2x - {\pi \over 4} \ne k\pi \Leftrightarrow x \ne {\pi \over 8} + {{k\pi } \over 2} \cr
& \Rightarrow D = R\backslash \left\{ {{\pi \over 8} + {{k\pi } \over 2};\,\,k \in Z} \right\} \cr
& c)\,\,y = {2 \over {\cos x - 1}} \cr
& DKXD:\,\,\cos x - 1 \ne 0 \Leftrightarrow \cos x \ne 0 \cr
& \Leftrightarrow x \ne {\pi \over 2} + k\pi \,\,\left( {k \in Z} \right) \cr
& \Rightarrow D = R\backslash \left\{ {{\pi \over 2} + k\pi ,\,\,k \in Z} \right\} \cr
& d)\,\,y = \sqrt {{{\sin x + 2} \over {\cos x + 1}}} \cr
& - 1 \le \sin x \le 1 \Leftrightarrow 1 \le \sin x + 2 \le 3 \cr
& \Rightarrow \sin x + 2 > 0\,\,\forall x \cr
& DKXD:\,\,\left\{ \matrix{
{{\sin x + 2} \over {\cos x + 1}} \ge 0 \hfill \cr
\cos x + 1 \ne 0 \hfill \cr} \right. \cr
& Do\,\,\sin x + 2 > 0\,\,\forall x \cr
& \Rightarrow \cos x + 1 > 0 \cr
& \Leftrightarrow \cos x > - 1 \cr
& \Leftrightarrow \cos x \ne - 1 \cr
& \Leftrightarrow x \ne \pi + k2\pi \,\,\left( {k \in Z} \right) \cr
& \Rightarrow D = R\backslash \left\{ {\pi + k2\pi ,\,\,k \in Z} \right\} \cr} $$
