The correct options are
A sin(α−β2)=1√2
B cos(α−β2)=−1√2
C Coordinates of orthocentre are (0,0)
Given : A(0,0),B(cosα,sinα) and C(cosβ,sinβ)
Finding the lengths of the triangle,
AB=√cos2α+sin2α=1 units
AC=√cos2β+sin2β=1 units
BC=√(cosα−cosβ)2+(sinα−sinβ)2⇒BC=√2−2(cosαcosβ+sinαsinβ)⇒BC=√2−2cos(α−β) units
As AC=AB, triangle is right angled isosceles triangle. So, hypotenuse is BC
Now, using Pythagoras theorem, we get
AB2+AC2=BC2
⇒2=2−2cos(α−β)
⇒cos(α−β)=0⇒α−β=(2n+1)π2,n∈Z∴sin(α−β2)=cos(α−β2)=±1√2
△ABC is right angled at A, so the coordinates of orthocentre are same as coordinates of A, i.e., (0,0)