Question

|2 tan x − 1| + |2 cot x − 1| = 2

Answer 1

\(VT=\left|2tanx-1\right|+\left|2cotx-1\right|\ge\left|2tanx+2cotx-2\right|\)

Mặt khác \(A=tanx+cotx\Rightarrow A^2=\left(tanx+cotx\right)^2=\left(tanx-cotx\right)^2+4\ge4\)

\(\Rightarrow\left[{}\begin{matrix}tanx+cotx\ge2\\tanx+cotx\le-2\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}2tanx+2cotx-2\ge2\\2tanx+2cotx-2\le-6\end{matrix}\right.\)

\(\Rightarrow\left|2tanx+2cotx-2\right|\ge2\)

Dấu "=" xảy ra khi và chỉ khi:

\(\left\{{}\begin{matrix}tanx=cotx\\tanx+cotx=2\\\left(2tanx-1\right)\left(2cotx-1\right)\ge0\end{matrix}\right.\) \(\Rightarrow x=\frac{\pi}{4}+k2\pi\)