Question

giải phương trình

\(\left(1+\cos x\right)\tan x=1+\cos x+\frac{2\sin x-1}{\cos x}\)

Answer 1

\(\left(1+cosx\right)tanx=1+cosx+\frac{2sinx-1}{cosx}\)

\(\Leftrightarrow sinx\left(1+cosx\right)=cos^2x+cosx+2sinx-1\)

\(\Leftrightarrow cos^2x+cosx+sinx-sinx.cosx-1=0\)

\(\Leftrightarrow cosx+sinx-sinx.cosx-sin^2x=0\)

\(\Leftrightarrow\left(cosx+sinx\right)\left(1-sinx\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx+sinx=0\\1-sinx=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\sqrt{2}sin\left(x+\frac{\Pi}{4}\right)=0\\sinx=1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\Pi}{4}+k\Pi\\x=\frac{\Pi}{2}+k2\Pi\end{matrix}\right.\)