Tính:
a) \(A = {\cos ^2}\frac{\pi }{8} + {\cos ^2}\frac{{3\pi }}{8} + {\cos ^2}\frac{{5\pi }}{8} + {\cos ^2}\frac{{7\pi }}{8}\)
b) \(B = \sin \frac{\pi }{5} + \sin \frac{{2\pi }}{5} + ... + \sin \frac{{9\pi }}{5}\) (có 9 số hạng)
c) \(C = \tan {1^o}{\rm{ }}.{\rm{ }}\tan {2^o}{\rm{ }}.{\rm{ }}\tan {3^o}.{\rm{ }}...{\rm{ }}{\rm{. }}\tan {89^o}\) (gồm 89 thừa số)
a) Sử dụng các công thức \(\cos \left( {\pi - x} \right) = - \cos x\), \(\cos \left( x \right) = \sin \left( {\frac{\pi }{2} - x} \right)\), \({\sin ^2}x + {\cos ^2}x = 1\).
b) Sử dụng công thức \(\sin \left( { - x} \right) = - \sin x\)
c) Sử dụng các công thức \(\tan x = \cot \left( {{{90}^o} - x} \right)\), \(\tan x.\cot x = 1\).
a) Ta có:
\(\cos \left( {\frac{{7\pi }}{8}} \right) = \cos \left( {\pi - \frac{\pi }{8}} \right) = - \cos \frac{\pi }{8}\)
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\(\cos \left( {\frac{{5\pi }}{8}} \right) = \cos \left( {\pi - \frac{{3\pi }}{8}} \right) = - \cos \frac{{3\pi }}{8}\)
\( \Rightarrow A = {\cos ^2}\frac{\pi }{8} + {\cos ^2}\frac{{3\pi }}{8} + {\cos ^2}\frac{{5\pi }}{8} + {\cos ^2}\frac{{7\pi }}{8}\)
\( = {\cos ^2}\frac{\pi }{8} + {\cos ^2}\frac{{3\pi }}{8} + {\cos ^2}\frac{{3\pi }}{8} + {\cos ^2}\frac{\pi }{8} = 2\left( {{{\cos }^2}\frac{\pi }{8} + {{\cos }^2}\frac{{3\pi }}{8}} \right)\)
Mặt khác, vì \(\cos \frac{{3\pi }}{8} = \sin \left( {\frac{\pi }{2} - \frac{{3\pi }}{8}} \right) = \sin \frac{\pi }{8}\)
Từ đó \(A = 2\left( {{{\cos }^2}\frac{\pi }{8} + {{\sin }^2}\frac{\pi }{8}} \right) = 2\).
b) Ta có: \(\sin \frac{{9\pi }}{5} = \sin \left( { - \frac{\pi }{5} + 2\pi } \right) = \sin \left( { - \frac{\pi }{5}} \right) = - \sin \frac{\pi }{5} \Rightarrow \sin \frac{{9\pi }}{5} + \sin \frac{\pi }{5} = 0\)
Tương tự ta có \(\sin \frac{{8\pi }}{5} + \sin \frac{{2\pi }}{5} = 0\), \(\sin \frac{{7\pi }}{5} + \sin \frac{{3\pi }}{5} = 0\), \(\sin \frac{{6\pi }}{5} + \sin \frac{{4\pi }}{5} = 0\)
Như vậy \(B = 0 + 0 + 0 + 0 + \sin \frac{{5\pi }}{5} = \sin \pi = 0\)
c) Ta có \(\tan {89^o} = \cot \left( {{{90}^o} - {{89}^o}} \right) = \cot {1^o}\), \(\tan {88^o} = \cot \left( {{{90}^o} - {{88}^o}} \right) = \cot {2^o}\),…
\(\tan {46^o} = \cot \left( {{{90}^o} - {{46}^o}} \right) = \cot {44^o}\).
Do đó \(C = \left( {\tan {1^o}.\tan {{89}^o}} \right)\left( {\tan {2^o}.\tan {{88}^o}} \right)...\left( {\tan {{44}^o}.\tan {{46}^o}} \right)\tan {45^o}\)
\( = \left( {\tan {1^o}.\cot {1^o}} \right)\left( {\tan {2^o}.\cot {2^o}} \right)...\left( {\tan {{44}^o}.\cot {{44}^o}} \right).1 = 1\)