Question

Упростите выражение: сos7β+cos3β+cos6β+сos2β=

Answer 1

$\cos( 7\beta ) + \cos(3 \beta ) + \cos( 6\beta ) + \cos( 2\beta ) = \\ = 2 \cos( \frac{ 7\beta + 3 \beta }{2} ) \cos( \frac{ 7\beta - 3 \beta }{2} ) + 2\cos( \frac{ 6\beta + 2 \beta }{2} ) \cos( \frac{ 6\beta - 2\beta }{2} ) = \\ = 2 \cos( 5\beta ) \cos(2 \beta ) + 2\cos( 4\beta ) \cos( 2\beta ) = \\ = 2 \cos( 2\beta ) \times ( \cos(5 \beta ) + \cos(4 \beta )) = \\ = 2 \cos( 2\beta ) \times 2 \cos( \frac{5 \beta + 4 \beta }{2} ) \cos( \frac{ 5\beta - 4 \beta }{2} ) = \\ = 4 \cos(2 \beta ) \cos( \frac{9 \beta }{2} ) \cos( \frac{ \ \beta }{2} )$