1.
ĐK: \(x\ne\dfrac{\pi}{2}+k\pi\)
\(1+sinx+cosx+tanx=0\)
\(\Leftrightarrow1+\dfrac{sinx}{cosx}+sinx+cosx=0\)
\(\Leftrightarrow\dfrac{sinx+cosx}{cosx}+sinx+cosx=0\)
\(\Leftrightarrow\left(sinx+cosx\right)\left(\dfrac{1}{cosx}+1\right)=0\)
\(\Leftrightarrow\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)\left(\dfrac{1}{cosx}+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(x+\dfrac{\pi}{4}\right)=0\\cosx=-1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{4}=k\pi\\x=\pi+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+k\pi\left(tm\right)\\x=\pi+k2\pi\left(tm\right)\end{matrix}\right.\)
2.
ĐK: \(x\ne\dfrac{k\pi}{2}\)
\(2tanx+cotx=2sin2x+\dfrac{1}{sin2x}\)
\(\Leftrightarrow2tanx-2sin2x+\dfrac{1}{tanx}-\dfrac{1}{sin2x}=0\)
\(\Leftrightarrow\left(tanx-sin2x\right)\left(2-\dfrac{1}{tanx.sin2x}\right)=0\)
\(\Leftrightarrow-tanx.cos2x.\dfrac{4cos^2x-1}{2cos^2x}=0\)
\(\Leftrightarrow cos2x.\left(2cos2x+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=0\\cos2x=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\x=\pm\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)
3.
ĐK: \(x\ne k\pi\)
\(1+cot2x=\dfrac{1-cos2x}{sin^2x}\)
\(\Leftrightarrow1+cot2x=\dfrac{2sin^2x}{sin^2x}\)
\(\Leftrightarrow cot2x=1\)
\(\Leftrightarrow2x=\dfrac{\pi}{4}+k\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{8}+\dfrac{k\pi}{2}\)
