(sin alpha+cos alpha)²=2-(sin alpha-cos alpha)² доказать тождество.

Question

(sin alpha+cos alpha)²=2-(sin alpha-cos alpha)² доказать тождество.

Answer 1

(sinα+cosα)²=2-(sinα-cosα)

1. (sinα+cosα)²=sin²α+2sinα*cosα+cos²α=
=(sin²α+cos²α)+2sinα*cosα=1+2sinα*cosα

2. 2-(sinα-cosα)²=2-(sin²α-2sinα*cosα+cos²α)=2-(sin²α+cos²α-2sinα*cosα)=
=2-(1-2sinα*cosα)=2-1-(-2sinα*cosα)=1+2sinα*cosα

3. 1+2sinα*cosα=1+2sinα*cosα
 ч. т. д.

Comments

Спасибо.