Докажите тождество 1+sina+cosa+tga=(1+cosa)(1+tga)

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Докажем данное тождество:

1 + sinα + cosα + tgα = 1 + (cosα + sinα) + tgα
= 1 + √2 sin(α + π/4) + tgα
= 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.