Bài 21 trang 214 sbt đại số 10

LG a

\(4{\cos ^4}a - 2\cos 2a - \dfrac{1}{2}\cos 4a\);

Lời giải chi tiết:

\(4{\cos ^4}a - 2\cos 2a - \dfrac{1}{2}\cos 4a\)

=\(4{\cos ^4}a - 2(2{\cos ^2}a - 1)\) \( - \dfrac{1}{2}(2{\cos ^2}2a - 1)\)

=\(4{\cos ^4}a - 4{\cos ^2}a + 2 \) \(- {(2{\cos ^2}a - 1)^2} + \dfrac{1}{2}\)

=\(4{\cos ^4}a - 4{\cos ^2}a + \dfrac{5}{2} \) \(- 4{\cos ^4}a + 4{\cos ^2}a - 1 \) \(= \dfrac{3}{2}\)

LG b

\({\sin ^2}a\left( {1 + \dfrac{1}{{\sin a}} + \cot a} \right)\left( {1 - \dfrac{1}{{\sin a}} + \cot a} \right)\);

Lời giải chi tiết:

\({\sin ^2}a(1 + \dfrac{1}{{\sin a}} + \cot a)(1 - \dfrac{1}{{\sin a}} + \cot a)\)

=\({\sin ^2}a\left[ {{{(1 + cota)}^2} - \dfrac{1}{{{{\sin }^2}a}}} \right] \) \(= {\sin ^2}a(1 + {\cot ^2}a + 2\cot a) - 1\)

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

LG c

\(\dfrac{{\cos 2a}}{{{{\cos }^4}a - {{\sin }^4}a}} - \dfrac{{{{\cos }^4}a + {{\sin }^4}a}}{{1 - \dfrac{1}{2}{{\sin }^2}2a}}\).

Lời giải chi tiết:

\(\dfrac{{\cos 2a}}{{{{\cos }^4}a - {{\sin }^4}a}} - \dfrac{{{{\cos }^4}a + {{\sin }^4}a}}{{1 - \dfrac{1}{2}{{\sin }^2}2a}}\)

=\(\dfrac{{{{\cos }^2}a - {{\sin }^2}a}}{{({{\cos }^2}a + {{\sin }^2}a)({{\cos }^2}a - {{\sin }^2}a)}} \) \(- \dfrac{{{{\cos }^4}a + {{\sin }^4}a}}{{1 - \dfrac{1}{2}{{(2\sin a\cos a)}^2}}}\)

\(\begin{array}{l}
= 1 - \dfrac{{{{\left( {{{\sin }^2}a + {{\cos }^2}a} \right)}^2} - 2{{\sin }^2}a{{\cos }^2}a}}{{1 - 2{{\sin }^2}a{{\cos }^2}a}}\\
= 1 - \dfrac{{1 - 2{{\sin }^2}a{{\cos }^2}a}}{{1 - 2{{\sin }^2}a{{\cos }^2}a}}\\
= 1 - 1 = 0
\end{array}\)