In a triangle ABC, if \(\frac{sin(A-B)}{sin(A+B)}=\frac{a^2-b^2}{a^2+

1.	In a triangle ABC, if \(\frac{sin(A-B)}{sin(A+B)}=\frac{a^2-b^2}{a^2+b^2},\) Prove that the triangle is either isosceles or right angled.
Answer» Given, \(\frac{sin(A-B)}{sin(A+B)}=\frac{a^2-b^2}{a^2+b^2}\) Applying componendo and dividendo \(\frac{sin(A-B)+sin(A+B)}{sin(A-B)-sin(A+B)} \)\(=\frac{a^2-b^2+a^2+b^2}{a^2-b^2-a^2-b^2}\) ⇒ \(-\frac{2\,sin\,A\,cos\,B}{2\,cos\,A\,sin\,B}=\frac{2a^2}{-2b^2}\) ⇒ \(\frac{sin\,A\,cos\,B}{cos\,A\,sin\,B}=\frac{a^2}{b^2}\) ⇒ b² sin A cos B = a² cos A sin B Putting sin A = ka, sin B = kb ⇒ b²ka cos B = a²kb cos A ⇒ b cos B = a cos A Applying cosine formula, \(b(\frac{a^2+c^2-b^2}{2ac})\) = \(a(\frac{b^2+c^2-b^2}{2bc})\) ⇒ b² (a²+ c² – b² ) = a² (b² + c² – a²) ⇒ a²b² + b²c²– b⁴ = a²b² + a²c² – a⁴ ⇒ b²c² – a²c² – b⁴ + a⁴ = 0 ⇒ c² (b² – a²) – (b² – a² ) (b²+ a² ) = 0 ⇒ (b² – a²) (c² – b² – a² ) = 0 ⇒ Either b² – a² = 0 or, c² – b² – a² = 0 ⇒ b = a or, c² = b² + a² ∴ Triangle is isosceles if b = a ∴ Triangle is right angled if c²= b² + a²

In a triangle ABC, if \(\frac{sin(A-B)}{sin(A+B)}=\frac{a^2-b^2}{a^2+b^2},\) Prove that the triangle is either isosceles or right angled.

Answer»

Given,

\(\frac{sin(A-B)}{sin(A+B)}=\frac{a^2-b^2}{a^2+b^2}\)

Applying componendo and dividendo

\(\frac{sin(A-B)+sin(A+B)}{sin(A-B)-sin(A+B)} \)\(=\frac{a^2-b^2+a^2+b^2}{a^2-b^2-a^2-b^2}\)

⇒ \(-\frac{2\,sin\,A\,cos\,B}{2\,cos\,A\,sin\,B}=\frac{2a^2}{-2b^2}\)

⇒ \(\frac{sin\,A\,cos\,B}{cos\,A\,sin\,B}=\frac{a^2}{b^2}\)

⇒ b² sin A cos B = a² cos A sin B

Putting sin A = ka, sin B = kb

⇒ b²ka cos B = a²kb cos A

⇒ b cos B = a cos A

Applying cosine formula,

\(b(\frac{a^2+c^2-b^2}{2ac})\) = \(a(\frac{b^2+c^2-b^2}{2bc})\)

⇒ b² (a²+ c² – b² ) = a² (b² + c² – a²)

⇒ a²b² + b²c²– b⁴ = a²b² + a²c² – a⁴

⇒ b²c² – a²c² – b⁴ + a⁴ = 0

⇒ c² (b² – a²) – (b² – a² ) (b²+ a² ) = 0

⇒ (b² – a²) (c² – b² – a² ) = 0

⇒ Either b² – a² = 0

or, c² – b² – a² = 0

⇒ b = a

or, c² = b² + a²

∴ Triangle is isosceles if b = a

∴ Triangle is right angled if c²= b² + a²