Q. In the ambiguous case, if b and A are given and  c 1 ,   c 2  are two possible values of the third side, prove that  c 1 - c 2 2 + c 1 + c 2 2 tan 2 A = 4 a 2

In a triangleCos A=b2+c2-a22bcAlso we knowc2-2b CosAc+b2-a2=0This is quadritic in c. Let the roots of this equation be c1 and c2. By formula of sum of roots and produt of rootsc1+c2=--2b cos A1=2b cos Ab=c1+c22 Cos Ac1.c2=b2-a21=b2-a2c1.c2=c1+c22 Cos A2-a2c1.c2=c1+c22 Cos A2-a2c1.c2=c1+c224 Cos2 A-a2c1.c2=c1+c224 sec2 A-a24c1.c2=c1+c22sec2 A-4a24c1.c2=c1+c221+tan2 A-4a24c1.c2=c1+c22+c1+c22tan2 A-4a2c1+c22-4c1.c2+c1+c22tan2 A=4a2c12+c22+2c1.c2-4c1.c2+c1+c22tan2 A=4a2c12+c22-2c1.c2+c1+c22tan2 A=4a2c1-c22+c1+c22tan2 A=4a2

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