Đơn giản biểu thức sau: a) sin^6a + cos^6a + 3sin^2a.cos^2a. b) sin^4a - cos^4a - (sina + cosa)(sina - cosa). c) cos^a + tan^2a*cos^2a

Đáp án:

\(\begin{array}{l}
a,\,\,\,\,1\\
b,\,\,\,\,0\\
c,\,\,\,\,1
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
a,\\
{\sin ^6}a + {\cos ^6}a + 3{\sin ^2}a.{\cos ^2}a\\
 = {\left( {{{\sin }^2}a} \right)^3} + {\left( {{{\cos }^2}a} \right)^3} + 3{\sin ^2}a.{\cos ^2}a\\
 = \left( {{{\sin }^2}a + {{\cos }^2}a} \right).\left[ {{{\left( {{{\sin }^2}a} \right)}^2} - {{\sin }^2}a.{{\cos }^2}a + {{\left( {{{\cos }^2}a} \right)}^2}} \right] + 3{\sin ^2}a.{\cos ^2}a\\
 = 1.\left[ {{{\left( {{{\sin }^2}a} \right)}^2} - {{\sin }^2}a.{{\cos }^2}a + {{\left( {{{\cos }^2}a} \right)}^2}} \right] + 3{\sin ^2}a.{\cos ^2}a\\
 = {\left( {{{\sin }^2}a} \right)^2} - {\sin ^2}a.{\cos ^2}a + {\left( {{{\cos }^2}a} \right)^2} + 3{\sin ^2}a.{\cos ^2}a\\
 = {\left( {{{\sin }^2}a} \right)^2} + 2.{\sin ^2}a.{\cos ^2}a + {\left( {{{\cos }^2}a} \right)^2}\\
 = {\left( {{{\sin }^2}a + {{\cos }^2}a} \right)^2}\\
 = {1^2}\\
 = 1\\
b,\\
{\sin ^4}a - {\cos ^4}a - \left( {\sin a + \cos a} \right).\left( {\sin a - \cos a} \right)\\
 = {\left( {{{\sin }^2}a} \right)^2} - {\left( {{{\cos }^2}a} \right)^2} - \left( {{{\sin }^2}a - {{\cos }^2}a} \right)\\
 = \left( {{{\sin }^2}a - {{\cos }^2}a} \right).\left( {{{\sin }^2}a + {{\cos }^2}a} \right) - \left( {{{\sin }^2}a - {{\cos }^2}a} \right)\\
 = \left( {{{\sin }^2}a - {{\cos }^2}a} \right).1 - \left( {{{\sin }^2}a - {{\cos }^2}a} \right)\\
 = \left( {{{\sin }^2}a - {{\cos }^2}a} \right) - \left( {{{\sin }^2}a - {{\cos }^2}a} \right)\\
 = 0\\
c,\\
{\cos ^2}a + {\tan ^2}a.{\cos ^2}a\\
 = {\cos ^2}a + {\left( {\dfrac{{\sin a}}{{\cos a}}} \right)^2}.{\cos ^2}a\\
 = {\cos ^2}a + \dfrac{{{{\sin }^2}a}}{{{{\cos }^2}a}}.{\cos ^2}a\\
 = {\cos ^2}a + {\sin ^2}a\\
 = 1
\end{array}\)