In any triangle. `if(a^2-b^2)/(a^2+b^2)=("sin"(A-B))/("sin"(A+B))` , then prove that the triangle is either right angled or isosceles.

Explanation

`(a^(2) - b^(2))/(a^(2) + b^(2)) = (sin (A - B))/(sin (A + B))`
or `(4R^(2) sin^(2) A - 4R^(2) sin^(2) B)/(4R^(2) sin^(2) A + 4R^(2) sin^(2) B) = (sin (A - B))/(sin (A + B))` (Using Sine Rule)
or `(sin (A + B) sin (A- B))/(sin^(2) A + sin^(2)B) = (sin(A - B))/(sin(A + B))`
`rArr sin(A - B) = 0 " or " (sin (pi C))/(sin^(2) A + sin^(2) B) = (1)/(sin (pi - C))`
or `A = B " or " sin^(2) C = sin^(2) A + sin^(2) B`
or `A = B " or " c^(2) = a^(2) + b^(2)` [from the sine rule]
Therefore, the triangle is isosceles or right angled