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)