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in triangle ABC,sin^3 Acos^3 (B-C)+sin^3 B cos^3 (C-A)+sin^3C cos^3 (A-B)=

in triangle ABC,sin^3 Acos^3 (B-C)+sin^3 B cos^3 (C-A)+sin^3C cos^3 (A-B)= 

Grade:10

3 Answers

Anoopam Mishra
126 Points
6 years ago
sin^3(A)cos^3(B-C) = sin^3(\pi - A)cos^3(B-C) = sin^3(B+C)cos^3(B-C) = (cos^2B - sin^2C)^3
So the given expression simplifies to
(cos^2B-sin^2C)^3 + (cos^2C-sin^2A)^3 + (cos^2A-sin^2B)^3
a^3 + b^3 + c^3 = 3abc \ , \ if \ a+b+c=0
3(cos^2B-sin^2C)(cos^2C-sin^2A)(cos^2A-sin^2B)
Lab Bhattacharjee
121 Points
6 years ago
\text{HINT:}\\ \cos^2x-\sin^2y=\cos(x-y)\cos(x+y) \\ 2\sin(x+y)\cos(x-y)=\sin2x+\sin2y
Raghu Vamshi Hemadri
72 Points
6 years ago
sin^3(A)cos^3(B-C) = sin^3(\pi - A)cos^3(B-C) = sin^3(B+C)cos^3(B-C) = (cos^2B - sin^2C)^3
So the given expression simplifies to
(cos^2B-sin^2C)^3 + (cos^2C-sin^2A)^3 + (cos^2A-sin^2B)^3
a^3 + b^3 + c^3 = 3abc \ , \ if \ a+b+c=0
 
3(cos^2B-sin^2C)(cos^2C-sin^2A)(cos^2A-sin^2B)

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