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prove that cos 2 A+cos 2 B+cos 2 C=1-2cosA.cosB.cocC please solve step by step?

prove that cos2A+cos2B+cos2C=1-2cosA.cosB.cocC please solve step by step?

Grade:11

1 Answers

Arun
25750 Points
6 years ago
cosC = cos(180−(A+B)) 
........ = −cos(A+B) 
........ = −(cosA cosB − sinA sinB) 
........ = sinA sinB − cosA cosB 

LHS = cos²A + cos²B + cos²C 
....... = cos²A + cos²B + (sinA sinB − cosA cosB)² 
....... = cos²A + cos²B + sin²A sin²B − 2 sinA sinB cosA cosB + cos²A cos²B 
....... = cos²A + cos²B + (1−cos²A) (1−cos²B) − 2 sinA sinB cosA cosB + cos²A cos²B 
....... = cos²A + cos²B + 1 − cos²A − cos²B + cos²A cos²B − 2 sinA sinB cosA cosB + cos²A cos²B 
....... = 1 − 2 sinA sinB cosA cosB + 2cos²A cos²B 
....... = 1 − 2 cosA cosB (sinA sinB − cosA cosB) 
....... = 1 − 2 cosA cosB cosC 
....... = RHS

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