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PROVE THAT : COS 2A – COS 2B + COS 2C = -1 -4 COSA COSB COSC

PROVE THAT :
COS 2A – COS 2B + COS 2C = -1 -4 COSA COSB COSC

Grade:11

1 Answers

Arun
25763 Points
4 years ago
cosC = cos(180−(A+B)) 
........ = −cos(A+B) 
........ = −(cosA cosB − sinA sinB) 
........ = sinA sinB − cosA cosB 

LHS = 2(cos²A + cos²B + cos²C) -3 
....... =2 [cos²A + cos²B + (sinA sinB − cosA cosB)²] -3 
....... = 2[cos²A + cos²B + sin²A sin²B − 2 sinA sinB cosA cosB + cos²A cos²B ] -3
....... = 2 [cos²A + cos²B + (1−cos²A) (1−cos²B) − 2 sinA sinB cosA cosB + cos²A cos²B] -3
....... = 2[cos²A + cos²B + 1 − cos²A − cos²B + cos²A cos²B − 2 sinA sinB cosA cosB + cos²A cos²B ] -3
....... = 2 − 4 sinA sinB cosA cosB + 4cos²A cos²B -3 
....... = -1 − 4 cosA cosB (sinA sinB − cosA cosB) 
....... = -1 − 4 cosA cosB cosC 
....... = RHS
 
 
 
Regards
Arun (askIITians forum expert)

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