ĐK: \(x\ne\pi+k2\pi;x\ne\dfrac{\pi}{2}+k\pi\)
\(tanx+\dfrac{sinx+cosx}{cosx+1}+1=0\)
\(\Leftrightarrow\dfrac{sinx}{cosx}+1+\dfrac{sinx+cosx}{cosx+1}=0\)
\(\Leftrightarrow\dfrac{sinx+cosx}{cosx}+\dfrac{sinx+cosx}{cosx+1}=0\)
\(\Leftrightarrow\left(\dfrac{1}{cosx}+\dfrac{1}{cosx+1}\right)\left(sinx+cosx\right)=0\)
\(\Leftrightarrow\dfrac{2cosx+1}{cos^2x+cosx}.sin\left(x+\dfrac{\pi}{4}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{2cosx+1}{cos^2x+cosx}=0\\sin\left(x+\dfrac{\pi}{4}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=-\dfrac{1}{2}\\sin\left(x+\dfrac{\pi}{4}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{2\pi}{3}+k2\pi\\x=-\dfrac{\pi}{4}+k\pi\end{matrix}\right.\)
