Câu 1
Có thể làm theo cách sau:
Theo giả thiết đã cho thì \alpha+\beta+\gamma=k\pi => sin(\alpha+\beta+\gamma)=0 và cos\alpha.cos\beta.cos\gamma
=>\frac{sin(\alpha+\beta+\gamma)}{cos\alpha.cos\beta.cos\gamma}=0 (1)
Ta lại có: sin(\alpha+\beta+\gamma)=sin(\alpha+\beta).cos\gamma+cos(\alpha+\beta).sin\gamma
=sin\alpha.cos\beta.cos\gamma+sin\beta.cos\alpha.cos\gamma+sin\gamma.cos\alpha.cos\beta-sin\alpha.sin\beta.sin\gamma (2)
Từ (1),(2) suy ra:
\frac{sin\alpha.cos\beta.cos\gamma+sin\beta.cos\alpha.cos\gamma+sin\gamma.cos\alpha.cos\beta-sin\alpha.sin\beta.sin\gamma}{cos\alpha.cos\beta.cos\gamma}=0
<=>tan\alpha+tan\beta+tan\gamma-tan\alpha.tan\beta.tan\gamma=0
<=>tan\alpha+tan\beta+tan\gamma=tan\alpha.tan\beta.tan\gamma (đpcm)