Statement 1: If `A` is an orthogonal matrix of order 2, then `|A|=+-1.` Statement 2: Every two-rowed real orthogonal matrix is of any one of the forms `(costheta-s inthetas inthetacostheta)or(costhetas inthetas intheta-costheta)dot`

Explanation

Let `A={:[(a_(1),a_(2)),(b_(1),b_(2))]:}` be a 2-rowed real orthogonal matrix.
From `A^(T)A={:[(a_(1),b_(1)),(a_(2),b_(2))]:}{:[(a_(1),a_(2)),(b_(1),b_(2))]:}`
`={:[(a_(1)^(2)+b_(1)^(2),a_(1)a_(2)+b_(1)b_(2)),(a_(1)a_(2)+b_(1)b_(2),a_(2)^(2)+b_(2)^(2))]:}=l={:[(1,0),(0,1)]:}`
`a_(1)^(2)+b_(1)^(2)=1,a_(1)a_(2)+b_(1)b_(2)=0,a_(1)^(2)+b_(2)^(2)=1`
As `a_(1),b_(1),a_(2),b_(2)inR`, from the 1st and 3rd relations it follows that `a_(i),b_(i)in[-1,-1]` for i=1,2
`{:("Accordingly let",a_(1)=costheta,a_(2)=cosphi),("so that",b_(1)=pmsintheta,b_(2)=pmsinphi):}`
`a_(1)a_(2)+b_(1)b_(2)=0` then transforms to
`cos(theta-phi)=0orcos(phi+theta)=0` according as we take same or different signs in the equations for `b_(1)` and `b_(2)`
Considering all combination of signs the following four possibilities emerge.
`{:[(costheta,-sintheta),(sintheta,costheta)]:},{:[(costheta,-sintheta),(-sintheta,-costheta)]:}`
`{:[(costheta,sintheta),(sintheta,-costheta)]:},{:[(costheta,sintheta),(-sintheta,costheta)]:}`
Changing `theta` to `-theta`, we observe that first and 2nd matrices respectively coincides with 4th and 3rd matrices, so that we have only two types of 2-rowed real orthogonal matrices, viz
`{:[(costheta,-sintheta),(sintheta,costheta)]:},{:[(costheta,sintheta),(sintheta,-costheta)]:},thetainR`