Cho góc alpha thoả mãn 180 độ < alpha < 270 độ. Rút gọn biểu thức: M = căn bậc hai (1 + cotα) sin^2 alpha + (1 + tan alpha) cos^2alpha.

\[\begin{array}{l} \Leftrightarrow M = \sqrt {\left( {1 + \frac{{\cos \alpha }}{{\sin \alpha }}} \right){{\sin }^2}\alpha  + \left( {1 + \frac{{\sin \alpha }}{{\cos \alpha }}} \right){{\cos }^2}\alpha } \\ \Leftrightarrow M = \sqrt {\frac{{\sin \alpha  + \cos \alpha }}{{\sin \alpha }}.{{\sin }^2}\alpha  + \frac{{\sin \alpha  + \cos \alpha }}{{\cos \alpha }}.{{\cos }^2}\alpha } \\ \Leftrightarrow M = \sqrt {\left( {\sin \alpha  + \cos \alpha } \right)(\sin \alpha  + \cos \alpha )}  = \left| {\sin \alpha  + \cos \alpha } \right|\end{array}\]

Do \({180^ \circ } < \alpha  < {270^ \circ } \Rightarrow \sin \alpha \,\,\& \,\,\cos \alpha  < 0\)

 Suy ra \(A =  - \left( {\sin \alpha  + \cos \alpha } \right)\).