Question

Докажите справедливость равенсва​

Answer 1

$1)4Cos(\frac{\pi }{12}+\frac{\alpha }{2}) *Cos(\frac{\pi }{12}-\frac{\alpha }{2})=\\\\=\frac{4(Cos(\frac{\pi }{12}+\frac{\alpha }{2}-\frac{\pi }{12}+\frac{\alpha }{2})+Cos(\frac{\pi }{12}+\frac{\alpha }{2}+\frac{\pi }{12}-\frac{\alpha }{2}))}{2}=\frac{4(Cos\alpha+Cos\frac{\pi }{6})}{2} =2(Cos\alpha+\frac{\sqrt{3} }{2})=\\\\=\boxed{\sqrt{3}+2Cos\alpha}$

$2)2\sqrt{2}Sin(\frac{\pi }{8}-\frac{\alpha }{2})*Cos(\frac{\pi }{8}+\frac{\alpha }{2})=\\\\=2\sqrt{2}*\frac{Sin(\frac{\pi }{8}-\frac{\alpha }{2}-\frac{\pi }{8}-\frac{\alpha }{2})+Sin(\frac{\pi }{8}-\frac{\alpha }{2}+\frac{\pi }{8}+\frac{\alpha }{2})}{2}=\sqrt{2}*(Sin(-\alpha)+Sin\frac{\pi }{4})=\\\\=\sqrt{2}*(\frac{1}{\sqrt{2}}-Sin\alpha)=\boxed{1-\sqrt{2}Sin\alpha }$