Știm formulele:
sin 2x=2sinx*cosx
a/sinA=b/sinB=c/sinC, de unde rezulta:
a*sinB=b*sinA; c*sinB=b*sinC
cosB=(a²+c²-b²)/(2ac)
cosC=(a²+b²-c²)/(2ab)
(ab*sinC)/2=(ac*sinB)/2=(bc*sinA)/2=S
Deci,
c²* sin2B +b²*sin2C =
=2c²* sinB*cosB+2b²* sinC*cosC
=2c²* sinB*cosB+2 b*c*sinB*cosC; (l-am inlocuit pe b*sinC cu c*sinB)
=2c*sinB( c* cosB+b*cosC); (am dat factor comun pe 2c*sinB)
=2c*sinB[ c*(a²+c²-b²)/(2ac) +b*(a²+b²-c²)/(2ab)]; (am inlocuit cosB si cosC)
=2c*sinB[ (a²+c²-b²+ a²+b²-c²)/(2a)
= 2c*sinB*2a²/(2a)
=4[(ac*sinB)/2]
=4S
