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A,B,C,D be postive angles such that A+B+C+D = 180, show that sinA.sinB+sinC.sinD = sin(A+C).sin(B+D)

A,B,C,D be postive angles such that 
A+B+C+D = 180, show that
 
sinA.sinB+sinC.sinD = sin(A+C).sin(B+D)

Grade:12

1 Answers

Vikas TU
14149 Points
7 years ago
Taking L.H.S
sinA.sinB+sinC.sinD = > 2sinA.sinB+ 2sinC.sinD 
=> cos(A-B) – cos(A+B) +  cos(C-D) – cos(C+D) 
-2sin(A-B+A+B)/2sin(A-B-A-B)/2  +  2sin(C-D+C+D)/2sin(C-D-C-D)/2
=> -2sinA

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