Question

Доведіть тотожність!!!!
Дам 30 балів

Answer 1

$sin(a+\beta )\cdot cos(a-\beta )=\\\\=(sina\cdot cos\beta +cosa\cdot sin\beta )(cosa\cdot cos\beta +sina\cdot sin\beta )=\\\\=sina\cdot cosa\cdot cos^2\beta +sin^2a\cdot sin\beta \cdot cos\beta +cos^2a\cdot sin\beta \cdot cos\beta +sin^2\beta \cdot sina\cdot cosa=\\\\=sina\cdot cosa\cdot (\underbrace{cos^2\beta +sin^2\beta }_{1})+sin\beta \cdot cos\beta \cdot (\underbrace{sin^2a+cos^2a}_{1})=\\\\=sina\cdot cosa+sin\beta \cdot cos\beta$

Или:

$\boxed{\ sinx\cdot cosxy=\dfrac{1}{2}\cdot \Big(\, sin(x+y)+sin(x-y)\, \Big)\ }\\\\\\sin(a+\beta )\cdot cos(a-\beta )=\dfrac{1}{2}\cdot \Big(\, sin(a+\beta +a-\beta )+sin(a+\beta -a+\beta )\Big)=\\\\\\=\dfrac{1}{2}\cdot (\, sin2a+sin2\beta \, )=\dfrac{1}{2}\, sin2a +\dfrac{1}{2}sin2\beta =\\\\\\=\dfrac{1}{2}\cdot 2sina\cdot cosa+\dfrac{1}{2}\cdot 2sin\beta \cdot cos\beta =sina\cdot cosa+sin\beta \cdot cos\beta$