Prove that `(a sin(B-C))/(b^(2)-c^(2))=(b sin(C-A))/(c^(2)-a^(2))=(c sin(A-B))/(a^(2)-b^(2))`

In any DeltaABC , prove that, (b^(2)-c^(2))/a^(2)sin2A+(c^(2)-a^(2))/b^(2)sin2B+(a^(2)-b^(2))/c^(2)sin2C=0 .

In any DeltaABC , prove that a^(3)sin(B-C)cosec^(2)A+b^(3)sin(C-A)cosec^(2)B+c^(3)sin(A-B)cosec^(2)C=0

Prove that (a^2sin(B-C))/(sinb+sinC)+(b^2"sin"(C-A))/(sinC+sinA)+(c^2"sin"(A-B))/(sinA+sinB)=0

In a Delta A B C , prove that: ((b^2-c^2)/(a^2))sin2A+((c^2-a^2)/(b^2))sin2B+((a^2-b^2)/(c^2))sin2C=0

If A+B+C=pi , prove that : (sin ((B+C)/(2)) + sin ((C+A)/(2)) + sin( (A+B)/(2) )) equals (4cos ((pi-A)/(4)) cos( (pi-B)/(4)) cos((pi-C)/(4))) .

In the triangle ABC , in which C is the right angle, prove that : sin 2 A =(2 ab )/(c ^(2)) , cos 2 A= ( b ^(2) - a ^(2))/( c ^(2)), sin ""(A)/(2) = sqrt ((c -b)/( 2c)) , cos ""(A)/(2)= sqrt ((c +b)/( 2c)).

In any triangle A B C , prove that: (a^2sin(B-C))/(sinB+ sin C)+(b^2sin(C-A))/(sinC+ sin A)+(c^2sin(A-B))/(sinA+ sin B)=0

In any triangle A B C , prove that following: \ (a^2"sin"(B-C))/(sin A)+(b^2"sin"(C-A))/(sin B)+(c^2"sin"(A-B))/(sin C)=0

Prove that ((a+b+c)(b+c-a)(c+a-b)(a+b-c))/(4b^2c^2)=sin^2A

Prove that: (sin(A-C)+2sinA+sin(A+C))/(sin(B-C)+2sinB+sin(B+C))=(sinA)/(sinB)