JEE Main & Advanced Mathematics Triangles & Properties of Triangle Question Bank Self Evaluation Test - Properties of Triangles and Height & Dstances

A, B, C are the angles of a triangle, then \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C-2\cos A\cos B\cos C=\]

A) 1

B) 2

C) 3

D) 4

Correct Answer: B

Solution :

[b] \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C\]

\[=1-{{\cos }^{2}}A+1-{{\cos }^{2}}B+{{\sin }^{2}}C\]

\[=2-{{\cos }^{2}}A-\cos (B+C)cos(B-C)\]

\[=2-\cos A[cosA-cos(B-C)]\]

\[=2-\cos A[-cos(B+C)-cos(B-C)]\]

\[=2+\cos A.2\cos B\cos C\]

\[\therefore {{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C-2\cos A\cos B\cos C=2\]