Let α and β be nonzero real numbers such that  2(cosβ-cosα)+cosαcosβ=1 . Then which of the following is/are true?

2cosβ−cosα+cosαcosβ=1⇒cosβ=1+2cosα2+cosα⇒1−y21+y2=1+21−x21+x22+1−x21+x2  x=tanα/2,y=tanβ/2⇒1−y21+y2=3−x23+x2⇒3y2=x23tanβ2+tanα23tanβ2−tanα2=0