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If a, b and A are given in a triangle an...

If a, b and A are given in a triangle and `c_(1), c_(2)` are possible values of the third side, then prove that `c_(1)^(2) + c_(2)^(2) - 2c_(1) c_(2) cos 2A = 4a^(2) cos^(2)A`

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We have `cos A = (b^(2) + c^(2) - a^(2))/(2bc)`
`rArr c^(2) - 2bc cos A + b^(2) - a^(2) = 0`
The equation which is equadratic in 'c'
`:. C_(1) + c_(2) = 2b cos A and c_(1) c_(2) = b^(2) - a^(2)` ..(i)
`:. C_(1)^(2) + c_(2)^(2) - 2c_(1) c_(2) cos 2A`
`= (c_(1) + c_(2))^(2) - 2c_(1) c_(2) - 2c_(1) c_(2) cos 2A` [using (i)]
`= (c_(1) + c_(2))^(2) - 2c_(1) c_(2) (1 + cos 2A)`
`= 4b^(2) cos^(2) A - 2 (b^(2) -a^(2)) 2 cos^(2) A`
`= 4 a^(2) cos^(2) A`
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