Question

(1+sin a+cos a)(sin a+cos a-1) 15 баллов

Answer 1

Ответ:

$sin2\alpha$

Объяснение:

$(1+sin\alpha+cos\alpha)(sin\alpha+cos\alpha-1)=(sin\alpha+cos\alpha+1)(sin\alpha+cos\alpha-1)=$

$=(sin\alpha+cos\alpha)^{2}-1^{2}=sin^{2}\alpha+2 \cdot sin\alpha \cdot cos\alpha+cos^{2}\alpha-1=2 \cdot sin\alpha \cdot cos\alpha+$

$+sin^{2}\alpha+cos^{2}\alpha-1=sin2\alpha+1-1=sin2\alpha;$

Answer 2

Alright, let's simplify the expression:

$(1+\sin a + \cos a)(\sin a + \cos a - 1)$

First, distribute the terms:

$= (1+\sin a + \cos a) \cdot \sin a + (1+\sin a + \cos a) \cdot \cos a - (1+\sin a + \cos a)$

Now, simplify each part:

$= \sin a + \sin^2 a + \sin a \cos a + \cos a \sin a + \cos^2 a + \cos a - 1 - \sin a - \cos a$

Combine like terms:

$= \sin^2 a + 2\sin a \cos a + \cos^2 a - 1$

Remember the trigonometric identity $\sin^2 a + \cos^2 a = 1$:

$= 2\sin a \cos a$

So, the simplified expression is $2\sin a \cos a$. If you have any other questions or need further clarification, feel free to ask!