Given:
[tex]5\tan ^2\beta+5\tan \beta=0[/tex][tex]5\tan \beta(\tan \beta+1)=0[/tex][tex]\begin{gathered} 5\tan \beta=0 \\ \tan \beta=0 \\ \beta=n\pi \end{gathered}[/tex][tex]\begin{gathered} \tan \beta+1=0 \\ \tan \beta=-1 \\ \beta=\frac{3\pi}{4}+n\pi \end{gathered}[/tex][tex]\beta=n\pi,\frac{3\pi}{4}+n\pi\text{ , where n is any integer}[/tex]
When n=0,
[tex]\beta=0,\frac{3\pi}{4}[/tex]
When n= -1
[tex]\beta=-\pi,-\frac{\pi}{4}[/tex]
When n= 1
[tex]\beta=\pi,\frac{7\pi}{4}\text{ }\Rightarrow\text{ Not valid because }-\pi\le\beta<\pi[/tex]
Therefore the final answer is
[tex]\beta=-\pi,-\frac{\pi}{4},0,\frac{3\pi}{4}[/tex]
