Half Angle Formula of `sin (A/2) ,Sin (B/2) ,sin(C/2)`

If A, B, C are angle of a triangle ABC, then the value of the determinant |(sin (A/2), sin (B/2), sin (C/2)), (sin(A+B+C), sin(B/2), cos(A/2)), (cos((A+B+C)/2), tan(A+B+C), sin (C/2))| is less than or equal to

If A,B,C and D are angles of quadrilateral and sin(A)/(2)sin(B)/(2)sin(C)/(2)sin(D)/(2)=(1)/(4) , prove that A=B=C=D= pi//2

If A,B,C and D are angles of quadrilateral and sin(A)/(2)sin(B)/(2)sin(C)/(2)sin(D)/(2)=(1)/(4) , prove that A=B=C=D= pi//2

If A,B,C and D are angles of quadrilateral and sin(A)/(2)sin(B)/(2)sin(C)/(2)sin(D)/(2)=(1)/(16) , prove that A=B=C=D=pi/2

If A,B,C and D are angles of quadrilateral and sin(A)/(2)sin(B)/(2)sin(C)/(2)sin(D)/(2)=(1)/(16) , prove that A=B=C=D=pi/2

If A+B+C= pi and (sin 2 A + sin 2B + sin 2 C)/(sin A + sin B + sin C ) = lamda sin ((A)/(2)) sin ((B)/(2)) sin ((C )/(2)) , then the value of lamda must be

A,B,C are three angles of a triangle and "sin" (A+c/2)= n "sin" C/2 ("sin"C/2)/("cos"(A-B)/2)=