prove that cos2A+cos2B+cos2C=1-2cosA.cosB.cocC please solve step by step?
kiran venkata , 6 Years ago
Grade 11
Trigonometry> prove that cos 2 A+cos 2 B+cos 2 C=1-2cos...
1 Answers
Arun
Last Activity: 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
........ = −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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