If A,B,C are the angles of a triangle, then the value of 1 - (sin^2 (A/2) + sin^2 (B/2) + sin^2 (C/2)) is ..?

A) 2 sin(A) sin(B) sin(C);


B) 2 sin(A/2) sin(B/2) sin(C/2);


C) 4 sin(A/2) sin(B/2) sin(C/2);


D) 4 sin(A) sin(B) sin(C);


Kindly explain your answer...

If A,B,C are the angles of a triangle, then the value of 1 - (sin^2 (A/2) + sin^2 (B/2) + sin^2 (C/2)) is ..?
1 - (sin^2 (A/2) + sin^2 (B/2) + sin^2 (C/2))


= 1 - 1/2(2sin^2 (A/2) + 2sin^2 (B/2) + 2sin^2 (C/2))


= 1 - 1/2(1- cos A + 1- cos B + 1- cos C)


= 1/2(cos A + cos B + cos C - 1)


= 1/2(2 cos(A+B)/2 cos(A-B)/2 - 2 sin^2 (C/2))


= cos(A+B)/2 cos(A-B)/2 - cos^2 ((A+B)/2)


= cos(A+B)/2 [cos(A-B)/2 - cos(A+B)/2]


= cos(A+B)/2 * 2 * sin A/2 * sin B/2


= 2 sin A/2 sin B/2 sin C/2;


Ans B.
Reply:Try some possibilities:





1) Let A=30, B=60, C=90. Then by the half angle formula, we get that





sin^2(30/2) = [1 - cos(30)]/2 = 1/2 - sqrt(3)/4





Also sin^2(60/2) = 1/4 and sin^2(90/2) = 1/2.





So the given expression equals sqrt(3)/4 - 1/4





Expression A) is 2 * 1/2 * sqrt(2)/2 = sqrt(2)/2





Epsression B) is 2 * sqrt(3)/2 * sqrt(2)/2 = sqrt(6)/2





Expression C) is sqrt(6), twice B).





Expression D) is sqrt(2), twice A).





None of these equals the value of the expression.

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