In ` A B C , sides b , c` and angle `B` are given such that `a` has two values `a_1and a_2dot` Then prove that `|a_1-a_2|=2sqrt(b^2-c^2sin^2B)`

Explanation

`cos B = (c^(2) + a^(2) - b^(2))/(2ca)`
or `a^(2) - (2c cos B) a + c^(2) - b^(2) = 0`
This equation has roots `a_(1) and a_(2)`
`rArr a_(1) + a_(2) = 2c cos B, a_(1) a_(2) = c^(2) - b^(2)`
`rArr (a_(1) -a_(2))^(2) = (a_(1) + a_(2))^(2) - 4a_(1) a_(2) = 4c^(2) cos^(2) B - 4(c^(2) - b^(2))`
`= 4b^(2) - 4c^(2) sin^(2) B = 4(b^(2) -c^(2) sin^(2) B)`
or `|a_(1) -a_(2)| = 2 sqrt(b^(2) - c^(2) sin^(2) B)`