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[tex]cos(\frac{23\pi }{12}) = cos(\frac{\pi }{6} +\frac{7\pi }{4} )=[/tex]
[tex]=cos(\frac{\pi }{6} )*cos(\frac{7\pi }{4}) -sin(\frac{\pi }{6} ) *sin(\frac{7\pi }{4} )=[/tex]
[tex]=cos(30\°)*cos(315\°)-sin(30\°)*sin(315\°)=[/tex]
[tex]=\frac{\sqrt{3} }{2} *\frac{\sqrt{2} }{2}-\frac{1}{2} } *(-\frac{\sqrt{2} }{2} )=[/tex]
[tex]= \frac{\sqrt{6} }{4} +\frac{\sqrt{2} }{4} =\\\\={\boxed{\mathb{\frac{\sqrt{6} +\sqrt{2} }{4} }}[/tex]
[tex]sin(\frac{\pi }{12} )= sin(\frac{\pi }{4} -\frac{\pi }{6} )=[/tex]
[tex]= sin(\frac{\pi }{4})* cos(\frac{\pi }{6} )- cos(\frac{\pi }{4} ) * sin(\frac{\pi }{6} ) =[/tex]
[tex]= sin(45\°)*cos(30\°)-cos(45\°)*sin(30\°) =[/tex]
[tex]=\frac{\sqrt{2} }{2} *\frac{\sqrt{3} }{2} -\frac{\sqrt{2} }{2} *\frac{1}{2} =[/tex]
[tex]=\frac{\sqrt{6} }{4} -\frac{\sqrt{2} }{4} =\\\\= {\boxed{\mathb{\frac{\sqrt{6}-\sqrt{2} }{4} }}[/tex]
[tex]cos(\frac{23\pi }{12} )*sin(\frac{\pi }{12} ) =[/tex]
[tex]=\frac{\sqrt{6} +\sqrt{2} }{4} *\frac{\sqrt{6} -\sqrt{2} }{4} =[/tex]
[tex]=\frac{(\sqrt{6} +\sqrt{2})*(\sqrt{6}-\sqrt{2})}{16} =[/tex]
[tex]=\frac{6-2}{16} =\frac{4}{16} ^{(4}= \\\\={\boxed{\boxed{\mathbf{\frac{1}{4} }}}[/tex]
