In a `Delta A B C ,` prove that: `((b^2-c^2)/(a^2))sin2A+((c^2-a^2)/(b^2))sin2B+((a^2-b^2)/(c^2))sin2C=0`

Explanation

Consider

`frac{b^{2}-c^{2}}{a^{2}} sin 2 A=frac{b^{2}-c^{2}}{a^{2}} cdot 2 sin A cos A `

`=frac{b^{2}-c^{2}}{a^{2}} cdot 2 frac{a}{k} cdot frac{b^{2}+c^{2}-a^{2}}{2 b c} `

`=frac{(b^{2}-c^{2})(b^{2}+c^{2}-a^{2})}{k a b c} `

`=frac{1}{k a b c}[b^{4}-c^{4}-a^{2} b^{2}+b^{2} a^{2}] `

Similarly

`frac{c^{2}-a^{2}}{b^{2}} sin 2 B=frac{1}{k a b c}[c^{4}-a^{4}-b^{2} c^{2}+b^{2} a^{2}] `

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