{1, 2} . {2, 3}

Solve[{
(ty1 - ly) (x1 - 
                lx) == (tx1 - lx) (y1 - 
                ly), 
(ty2 - y1) (x2 - 
                x1) == (tx2 - x1) (y2 - 
                y1), 
(x1 - lx) (tx1 - 
                  cx) + (y1 - ly) (ty1 - cy) == 
          0, 
(x2 - x1) (tx2 - cx) + (y2 - 
                  y1) (ty2 - cy) == 
          0, 
(r^2) == (((tx1 - cx))^2) + (((ty1 - cy))^2), 
(r^2) == (((tx2 - cx))^2) + (((ty2 - cy))^2)}, {tx1, tx2, ty1, ty2, cx, cy}]

(ty1 - ly) (x1 - 
              lx) == (tx1 - lx) (y1 - ly) /. {tx1 -> x1, 
        ty1 -> y1}

(ty2 - y1) (x2 - 
              x1) == (tx2 - x1) (y2 - y1) /. {tx2 -> x2, 
        ty2 -> y2}

Tan[ArcCos[q]] // Simplify

Tan[(1/2) ArcCos[q]] // FullSimplify

Tan[(1/2) θ] // FullSimplify

(1/2) (1 + Cos[a]) // FullSimplify

(1/2) (1 - Cos[a]) // FullSimplify

((1 - Cos[a])/(1 + Cos[a])) // FullSimplify

((1 - Cos[a])/(1 + Cos[a])) /. {a -> ArcCos[q]}

Tan[ArcCos[q]/2] - (Sqrt[((1 - q)/(1 + q))])

Plot[
      Tan[ArcCos[q]/2] -  (Sqrt[((1 - q)/(1 + q))]), {q, -1, 
        1}]

(Sqrt[((1 - (d/(a b)))/(1 + (d/(a b))))]) // 
      FullSimplify

(( a b - d)/(a b + d)) // Expand