Biểu thức A = cos^2alpha . cot^2alpha + 3cos^2alpha – cot^2alpha

Đáp án đúng là: C

Ta có: A = cos²α.cot²α + 3cos²α – cot²α +2sin² α

\( = {\cos ^2}\alpha .\frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} - \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + 2{\sin ^2}\alpha + 3{\cos ^2}\alpha \)

\( = {\cos ^2}\alpha .\frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} - \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + 2\left( {{{\sin }^2}\alpha + {{\cos }^2}\alpha } \right) + {\cos ^2}\alpha \)

\( = \frac{{{{\cos }^4}\alpha }}{{{{\sin }^2}\alpha }} - \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + 2 + {\cos ^2}\alpha \)

\( = \frac{{{{\cos }^4}\alpha - {{\cos }^2}\alpha + {{\sin }^2}\alpha .{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + 2\)

\( = \frac{{{{\cos }^2}\alpha ({{\sin }^2}\alpha + {{\cos }^2}\alpha ) - {{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + 2\)

\( = \frac{{{{\cos }^2}\alpha - {{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} + 2\)

= 2.