Question

Solve the following equations.
$tanx+sin(\pi +x)=2si{n}^2\frac{x}{2}.$

Answer 1

$tanx+sin(\pi +x)=2si{n}^{2}x/2$

$\Rightarrow \frac{sinx}{cosx}-sinx=2si{n}^{2}x/2$

$\Rightarrow \frac{sinx-sinxcosx}{cosx}=1-cosx$

$\Rightarrow sinx(1-cosx)=cosx(1-cosx)$

$\Rightarrow (1-cosx)(sinx-cosx)=0$

$\therefore cosx=1$ or $sinx-cosx=0$

$\Rightarrow x=2x\pi$ or $\Rightarrow tanx=1$

$nϵ$ integer $\Rightarrow x=m\pi +\pi /4$

$mϵ$ integer