Whether the given equation b2-c2-a2sin2A-c2-a2-b2sin2B-a2-b2-c2sin2C-1 -In any -ABC

The correct option is B FalseProve that (b^2 c^2)a^2sin 2A+(c^2 a^2)b^2sin 2B+(a^2 b^2)c^2sin 2C=1 By sine rule sin Aa=sin Bb=sin Cc=k cos A=b^2 ...

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Standard XII

Mathematics

Property 1

Whether the g...

Question

Whether the given equation $(\frac{b^{2} - c^{2}}{a^{2}}) s i n 2 A + (\frac{c^{2} - a^{2}}{b^{2}}) s i n 2 B + (\frac{a^{2} - b^{2}}{c^{2}}) s i n 2 C = 1$ ?(In any $Δ$ ABC )

True

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False

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Solution

The correct option is B False
Prove that $\frac{(b^{2} - c^{2})}{a^{2}} sin 2 A + \frac{(c^{2} - a^{2})}{b^{2}} sin 2 B + \frac{(a^{2} - b^{2})}{c^{2}} sin 2 C = 1$
By sine rule $\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c} = k$
$cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$
LHS
$\Rightarrow (\frac{b^{2} - c^{2}}{a^{2}}) 2 a k + (\frac{b^{2} + c^{2} - a^{2}}{2 b c}) + (\frac{c^{2} - a^{2}}{b^{2}}) 2 b k (\frac{a^{2} + c^{2} - b^{2}}{2 a c}) + (\frac{a^{2} - b^{2}}{c^{2}}) 2 c k (\frac{a^{2} + b^{2} - c^{2}}{2 a b})$
$= \frac{k}{a b c} (b^{2} - c^{2}) (b^{2} + c^{2} - a^{2}) + (c^{2} - a^{2}) (a^{2} + c^{2} - b^{2}) + (a^{2} - b^{2}) (a^{2} + b^{2} - c^{2})$
$= \frac{k}{a b c} (b^{2} - b^{2} / c^{2} - a^{2} / b^{2} - a^{2} c^{2} - c^{4} + a^{2} c^{2} + a^{2} c^{2} + c^{4} - b^{2} / c^{2} - a^{4} - a^{2} c^{2} + a^{2} b^{2} + a^{4} + a^{2} b^{2} - a^{2} c^{2} - a^{2} b^{2} - b^{4} + b^{2} c^{2})$
$= \frac{k}{a b c} (0)$
$= 0$
Hence $0 = 1$
LHS $\neq$ RHS
Hence not proved.

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Whether the given equation (b2−c2a2)sin2A+(c2−a2b2)sin2B+(a2−b2c2)sin2C=1 ?(In any ΔABC )

Whether the given equation $(\frac{b^{2} - c^{2}}{a^{2}}) s i n 2 A + (\frac{c^{2} - a^{2}}{b^{2}}) s i n 2 B + (\frac{a^{2} - b^{2}}{c^{2}}) s i n 2 C = 1$ ?(In any $Δ$ ABC )