Question

Упростите выражение: cos5a+cos6a+cos7a/sin5a+sin6a+sin7a

Answer 1

$\frac{cos5a+cos6a+cos7a}{sin5a+sin6a+sin7a} = \\ = \frac{(cos5a+cos7a)+cos6a}{(sin5a+sin7a)+sin6a} = \\ = \frac{2cos \frac{5a + 7a}{2}cos \frac{5a - 7a}{2} + cos6a}{2sin\frac{5a + 7a}{2}cos \frac{5a - 7a}{2} + sin6a} = \\ = \frac{2cos 6a \times cos a + cos6a}{2sin6a \times cos a+ sin6a} = \\ = \frac{cos 6a (2cos a + 1)}{sin6a( 2cos a+ 1)} = ctg6a$

Ответ: сtg 6a