[tex]\tan\beta+\cot\beta=2[/tex]
Recall that [tex]\cot\beta=\frac1{\tan\beta}[/tex]. So if [tex]\tan\beta\neq0[/tex], we can multiply both sides of the equation by [tex]\tan\beta[/tex] to get
[tex]\tan^2\beta+1=2\tan\beta\implies\tan^2\beta-2\tan\beta+1=0[/tex]
[tex]\implies(\tan\beta-1)^2=0[/tex]
[tex]\implies\tan\beta-1=0[/tex]
[tex]\implies\tan\beta=1[/tex]
[tex]\implies\boxed{\beta=45^\circ,225^\circ}[/tex]
