$$\eqalign{
& 1)\,\,y = {\sin ^2}x - \sin x + 2 \cr
& y = {\sin ^2}x - 2.\sin x.{1 \over 2} + {1 \over 4} + {7 \over 4} \cr
& y = {\left( {\sin x - {1 \over 2}} \right)^2} + {7 \over 4} \cr
& - 1 \le \sin x \le 1 \cr
& \Leftrightarrow - {3 \over 2} \le \sin x - {1 \over 2} \le {1 \over 2} \cr
& \Leftrightarrow 0 \le {\left( {\sin x - {1 \over 2}} \right)^2} \le {9 \over 4} \cr
& \Leftrightarrow {7 \over 4} \le {\left( {\sin x - {1 \over 2}} \right)^2} + {7 \over 4} \le 4 \cr
& \Rightarrow {\mathop{\rm miny}\nolimits} = {7 \over 4};\,\,{\mathop{\rm maxy}\nolimits} = 4 \cr
& 3)\,\,y = si{n^2}x - \cos x + 2 \cr
& y = 1 - {\cos ^2}x - \cos x + 2 \cr
& y = - {\cos ^2}x - \cos x + 3 \cr
& y = - \left( {{{\cos }^2}x + \cos x} \right) + 3 \cr
& y = - \left( {{{\cos }^2}x + 2\cos x.{1 \over 2} + {1 \over 4}} \right) + {1 \over 4} + 3 \cr
& y = - {\left( {\cos x + {1 \over 2}} \right)^2} + {{13} \over 4} \cr
& - 1 \le \cos x \le 1 \cr
& \Leftrightarrow - {1 \over 2} \le \cos x + {1 \over 2} \le {3 \over 2} \cr
& \Leftrightarrow 0 \le {\left( {\cos x + {1 \over 2}} \right)^2} \le {9 \over 4} \cr
& \Leftrightarrow 0 \ge - {\left( {\cos x + {1 \over 2}} \right)^2} \ge - {9 \over 4} \cr
& \Leftrightarrow {{13} \over 4} \ge - {\left( {\cos x + {1 \over 2}} \right)^2} + {{13} \over 4} \ge 1 \cr
& \Rightarrow {\mathop{\rm miny}\nolimits} = 1;\,\,maxy = {{13} \over 4} \cr} $$
$$\eqalign{
& 2)\,y = {\cos ^2}x - 2\cos x + 6 \cr
& y = {\cos ^2}x - 2\cos x + 1 + 5 \cr
& y = {\left( {\cos x - 1} \right)^2} + 5 \cr
& - 1 \le \cos x \le 1 \cr
& \Leftrightarrow - 2 \le \cos x - 1 \le 0 \cr
& \Leftrightarrow 0 \le {\left( {\cos x - 1} \right)^2} \le 4 \cr
& \Leftrightarrow 5 \le {\left( {\cos x - 1} \right)^2} + 5 \le 9 \cr
& \Rightarrow \min = 5;\,\,\max = 9 \cr} $$
