Xét tính đúng sai của các mệnh đề sau:

Đáp án: a) Đúng. b) Sai. c) Sai. d) Sai.

a) Đúng. \(A = \frac{{{{\cos }^3}a + {{\sin }^3}a}}{{\cos a + \sin a}} + \sin a \cdot \cos a\)

\( = \frac{{\left( {\cos a + \sin a} \right) \cdot \left( {{{\cos }^2}a - \cos a \cdot \sin a + {{\sin }^2}a} \right)}}{{\cos a + \sin a}} + \sin a \cdot \cos a\)

\( = {\cos ^2}a - \cos a \cdot \sin a + {\sin ^2}a + \sin a \cdot \cos a = 1\).

b) Sai. \(B = 2\left( {{{\sin }^6}a + {{\cos }^6}a} \right) - 3\left( {{{\sin }^4}a + {{\cos }^4}a} \right)\)

= \(2\left( {{{\sin }^2}a + {{\cos }^2}a} \right) \cdot \left( {{{\sin }^4}a - {{\sin }^2}a \cdot {{\cos }^2}a + {{\cos }^4}a} \right)\)

\( - 3 \cdot \left[ {{{\left( {{{\sin }^2}a + {{\cos }^2}a} \right)}^2} - 2{{\sin }^2}a \cdot {{\cos }^2}a} \right]\)

\( = 2 \cdot \left[ {1 - 3{{\sin }^2}a \cdot {{\cos }^2}a} \right] - 3\left( {1 - 2{{\sin }^2}a \cdot {{\cos }^2}a} \right)\)

= 2 – 6sin²acos²a – 3+ 6sin²acos²a = −1.

c) Sai. C = 8(cos⁸a – sin⁸a) – cos6a – 7cos2a

= 8(cos⁴a – sin⁴a)(sin⁴a + cos⁴a) – cos6a – 7cos2a

= 8(cos²a – sin²a)(sin²a + cos²a)[(sin²a + cos²a)² – 2sin²acos²a] – cos6a – 7cos2a

= 8(cos²a – sin²a)(1 − 2sin²acos²a) – cos6a – 7cos2a

\( = 8 \cdot \left( {\cos 2a} \right)\left( {1 - \frac{1}{2}{{\sin }^2}2a} \right) - \cos 6a - 7\cos 2a\)

\( = 8\cos 2a - 4\cos 2a \cdot {\sin ^2}2a - \cos 6a - 7\cos 2a = \cos 2a - 4\cos 2a\left( {1 - {{\cos }^2}2a} \right) - \cos 6a\)

\( = \cos 2a - 4\cos 2a + 4{\cos ^3}2a - \cos 6a = \left( {4{{\cos }^3}2a - 3\cos 2a} \right) - \cos 6a = \cos 6a - \cos 6a = 0\) d) Sai. D = cos²x – 2cosa∙cosx∙cos(a + x) + cos²(a + x).

= cos²x – [cos(a + x) + cos(a – x)]∙cos(a + x) + cos²(a + x).

= cos²x – cos²(a + x) – cos(a – x)cos(a + x) + cos²(a + x)

= cos²x – cos(a – x)cos(a + x)

= cos²x – (cosa∙cosx − sina∙sinx)( cosa∙cosx + sina∙sinx)

= cos²x – (cos²a∙cos²x) + (sin²asin²x) = cos²x(1 – cos²a) + sin²asin²x

= cos²x∙sin²a + sin²asin²x = sin²a(cos²x + sin²x) = sin²a.