Question

Алгебра, 10 класс
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Answer 1

$1)\log_{\frac{1}{\sqrt{2} } } \Big(Cos^{2}22,5^\circ-Sin^{2}22,5^\circ\Big)=\log_{\frac{1}{\sqrt{2}}} Cos(2\cdot 22,5^\circ)=\\\\=\log_{\frac{1}{\sqrt{2} } }Cos 45^\circ=\log_{\frac{1}{\sqrt{2}}}\frac{1}{\sqrt{2} }=\boxed1$

$2)\log_{\frac{\sqrt{3} }{2} }\Big(Cos\frac{\pi }{12}-Sin\frac{\pi }{12}\Big)+\log_{\frac{\sqrt{3} }{2} }\Big(Cos\frac{\pi }{12}+Sin\frac{\pi }{12}\Big)=\\\\=\log_{\frac{\sqrt{3} }{2} }\Big[(Cos\frac{\pi }{12}-Sin\frac{\pi }{12})\cdot (Cos\frac{\pi }{12}+Sin\frac{\pi }{12})\Big] =\log_{\frac{\sqrt{3} }{2} }\Big(Cos^{2}\frac{\pi }{12}-Sin^{2}\frac{\pi }{12}\Big)=$

$=\log_{\frac{\sqrt{3} }{2} }Cos(2\cdot \frac{\pi }{12})=\log_{\frac{\sqrt{3} }{2} }Cos\frac{\pi }{6} = \log_{\frac{\sqrt{3} }{2} }\frac{\sqrt{3} }{2}=\boxed1$