Question

упростите выражение, пожалуйста

Answer 1

$\displaystyle\bf\\Sin\alpha (Cos2\alpha +Cos4\alpha +Cos6\alpha +Cos8\alpha )=\\\\\\=Sin\alpha \cdot \Big[\Big(Cos2\alpha +Cos8\alpha \Big)+\Big(Cos4\alpha +Cos6\alpha \Big)\Big]=\\\\\\=Sin\alpha \cdot \Big(2Cos\frac{2\alpha +8\alpha }{2} Cos\frac{2\alpha -8\alpha }{2} +2Cos\frac{4\alpha +6\alpha }{2} Cos\frac{4\alpha -6\alpha }{2} \Big)=\\\\\\=Sin\alpha \cdot\Big(2Cos5\alpha Cos3\alpha +2Cos5\alpha Cos\alpha\Big)=2Sin\alpha Cos5\alpha \Big(Cos3\alpha +Cos\alpha \Big)=$

$\displaystyle\bf\\=2Sin\alpha Cos5\alpha \cdot 2Cos\frac{3\alpha +\alpha }{2}Cos\frac{3\alpha -\alpha }{2}=4Sin\alpha Cos5\alpha Cos2\alpha Cos\alpha = \\\\\\=\underbrace{2Sin\alpha Cos\alpha} _{Sin2\alpha }\cdot 2Cos5\alpha Cos2\alpha =\underbrace{2Sin2\alpha Cos2\alpha }_{Sin4\alpha }\cdot Cos5\alpha =Sin4\alpha Cos5\alpha$