Докажем данное тождество:
1 + sinα + cosα + tgα = 1 + (cosα + sinα) + tg= 1 + √2 sin(α + π/4) + tg= 1 + √2 sin(α + π/4) + sin(α) / cos(α= (cos(π/4) + sin(α)) / cos(α) + √2 sin(α + π/4= (cos(π/4) cos(α) + sin(α) cos(π/4)) / cos(α) + √2 sin(α + π/4= ((cos(π/4) sin(α) + sin(α) cos(π/4)) + cos(α) sin(α)) / cos(α) + √2 sin(α + π/4= (sin(α + π/4) + cos(α) sin(α)) / cos(α) + √2 sin(α + π/4= (sinα cos(π/4) + cosα sin(π/4)) / cos(α) + √2 sin(α + π/4= ((cos(α - π/4)) / cos(α) + √2 sin(α + π/4= (cos(π/4) / cos(α)) + √2 sin(α + π/4= (1 / cos(α)) + √2 sin(α + π/4= sec(α) + √2 sin(α + π/4= sec(α) + √2/2 (sinα + cosα= (1 + cosα) sec(α= (1 + cosα)(1 + tgα)
Тождество 1 + sinα + cosα + tgα = (1 + cosα)(1 + tgα) доказано.
Докажем данное тождество:
1 + sinα + cosα + tgα = 1 + (cosα + sinα) + tg
= 1 + √2 sin(α + π/4) + tg
= 1 + √2 sin(α + π/4) + sin(α) / cos(α
= (cos(π/4) + sin(α)) / cos(α) + √2 sin(α + π/4
= (cos(π/4) cos(α) + sin(α) cos(π/4)) / cos(α) + √2 sin(α + π/4
= ((cos(π/4) sin(α) + sin(α) cos(π/4)) + cos(α) sin(α)) / cos(α) + √2 sin(α + π/4
= (sin(α + π/4) + cos(α) sin(α)) / cos(α) + √2 sin(α + π/4
= (sinα cos(π/4) + cosα sin(π/4)) / cos(α) + √2 sin(α + π/4
= ((cos(α - π/4)) / cos(α) + √2 sin(α + π/4
= (cos(π/4) / cos(α)) + √2 sin(α + π/4
= (1 / cos(α)) + √2 sin(α + π/4
= sec(α) + √2 sin(α + π/4
= sec(α) + √2/2 (sinα + cosα
= (1 + cosα) sec(α
= (1 + cosα)(1 + tgα)
Тождество 1 + sinα + cosα + tgα = (1 + cosα)(1 + tgα) доказано.