Rút gọn các biểu thức (không dùng bảng số và máy tính)
a) \({\sin ^2}\left( {{{180}^0} - \alpha } \right) + {\tan ^2}\left( {{{180}^0} - \alpha } \right){\tan ^2}\left( {{{270}^0} + \alpha } \right) + \sin \left( {{{90}^0} + \alpha } \right)\cos \left( {\alpha - {{360}^0}} \right)\)
b) \(\frac{{\cos \left( {\alpha - {{90}^0}} \right)}}{{\sin \left( {{{180}^0} - \alpha } \right)}} + \frac{{\tan \left( {\alpha - {{180}^0}} \right)\cos \left( {{{180}^0} + \alpha } \right)\sin \left( {{{270}^0} + \alpha } \right)}}{{\tan \left( {{{270}^0} + \alpha } \right)}}\)
c) \(\frac{{\cos \left( { - {{288}^0}} \right)\cot {{72}^0}}}{{\tan \left( { - {{162}^0}} \right)\sin {{108}^0}}} - \tan {18^0}\)
d) \(\frac{{\sin {{20}^0}\sin {{30}^0}\sin {{40}^0}\sin {{50}^0}\sin {{60}^0}\sin {{70}^0}}}{{\cos {{10}^0}\cos {{50}^0}}}\)
a) \({\sin ^2}\left( {{{180}^0} - \alpha } \right) + {\tan ^2}\left( {{{180}^0} - \alpha } \right){\tan ^2}\left( {{{270}^0} + \alpha } \right) + \sin \left( {{{90}^0} + \alpha } \right)\cos \left( {\alpha - {{360}^0}} \right)\)
\( = {\sin ^2}\alpha + {\tan ^2}\alpha {\cot ^2}\alpha + {\cos ^2}\alpha = 2\)
b) \(\frac{{\cos \left( {\alpha - {{90}^0}} \right)}}{{\sin \left( {{{180}^0} - \alpha } \right)}} + \frac{{\tan \left( {\alpha - {{180}^0}} \right)\cos \left( {{{180}^0} + \alpha } \right)\sin \left( {{{270}^0} + \alpha } \right)}}{{\tan \left( {{{270}^0} + \alpha } \right)}}\)
\( = \frac{{\sin \alpha }}{{\sin \alpha }} + \frac{{\tan \alpha \left( { - \cos \alpha } \right)\left( { - \cos \alpha } \right)}}{{ - \cot \alpha }} = 1 - {\sin ^2}\alpha = {\cos ^2}\alpha \)
c) \(\frac{{\cos \left( { - {{288}^0}} \right)\cot {{72}^0}}}{{\tan \left( { - {{162}^0}} \right)\sin {{108}^0}}} - \tan {18^0}\)
\(\begin{array}{l}
= \frac{{\cos \left( {{{72}^0} - {{360}^0}} \right)\cot {{72}^0}}}{{\tan \left( {{{18}^0} - {{180}^0}} \right)\sin \left( {{{180}^0} - {{72}^0}} \right)}} - \tan {18^0}\\
= \frac{{\cos {{72}^0}\cot {{72}^0}}}{{\tan {{18}^0}\sin {{72}^0}}} - \tan {18^0}\\
= \frac{{{{\cot }^2}{{72}^0}}}{{\tan {{18}^0}}} - \tan {18^0} = \frac{{{{\tan }^2}{{18}^0}}}{{\tan {{18}^0}}} - \tan {18^0} = 0
\end{array}\)
d) Ta có \(\sin {70^0} = \cos {20^0},\sin {50^0} = \cos {40^0},\sin {40^0} = \cos {50^0}\). Vì vậy
\(\frac{{\sin {{20}^0}\sin {{30}^0}\sin {{40}^0}\sin {{50}^0}\sin {{60}^0}\sin {{70}^0}}}{{\cos {{10}^0}\cos {{50}^0}}}\)
\(\begin{array}{l}
\frac{{\frac{1}{2}.\frac{{\sqrt 3 }}{2}.\sin {{20}^0}\cos {{20}^0}\cos {{50}^0}\cos {{40}^0}}}{{\cos {{10}^0}\cos {{50}^0}}} = \frac{{\frac{1}{2}.\frac{{\sqrt 3 }}{4}.\sin {{40}^0}\cos {{40}^0}}}{{\cos {{10}^0}}}\\
= \frac{{\frac{{\sqrt 3 }}{{16}}\sin {{80}^0}}}{{\cos {{10}^0}}} = \frac{{\sqrt 3 }}{6}
\end{array}\)
-- Mod Toán 10