If (a^2-b^2)/(a^2+b^2)={sin(A-B)}/{sin(A+B)}then prove that the triangle is either a right angled triangle or an isosceles triangle.
By the sine rule, . So the left side of the above equation is equal to . Use the addition formulae on the right side of the equation, and it becomes Multiply out the fractions, do some cancelling and rearrangement, and you get . If either of the first two factors is zero then the triangle is right-angled. If then either (in which case the triangle is again right-angled), or and the triangle is isosceles.