Question

Nếu x+y+z=π/2

Cot x + cot y = 2cot z

Thì cot x . Coty = ?

Answer 1

\(x+y+z=\frac{\pi}{2}\Rightarrow x+y=\frac{\pi}{2}-z\)

\(\Rightarrow tan\left(x+y\right)=tan\left(\frac{\pi}{2}-z\right)=cotz\)

\(\Rightarrow\frac{tanx+tany}{1-tanx.tany}=cotz\)

Mà \(cotx+coty=2cotz\Rightarrow cotx+coty=\frac{2\left(tanx+tany\right)}{1-tanx.tany}\)

\(\Rightarrow\frac{1}{tanx}+\frac{1}{tany}=\frac{2\left(tanx+tany\right)}{1-tanx.tany}\Leftrightarrow\frac{tanx+tany}{tanx.tany}=\frac{2\left(tanx+tany\right)}{1-tanx.tany}\)

\(\Rightarrow\frac{1}{tanx.tany}=\frac{2}{1-tanx.tany}\Leftrightarrow1-tanx.tany=2tanx.tany\)

\(\Rightarrow tanx.tany=\frac{1}{3}\Rightarrow cotx.coty=3\)