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

Arun
25763 Points
3 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