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In DeltaABC, prove that: cos2A+cos2B-c...

In `DeltaABC`, prove that:
`cos2A+cos2B-cos2C=1-4sinAsinBcosC`

Text Solution

Verified by Experts

LHS `=cos2A+cos2B-cos2C`
`=2cos(A+B)cos(A-B)-(2cos^(2)C-1)`
`=2cos(180^(@)-C)cos(A-B)-2cos^(2)C+1` `(therefore A+B+C=180^(@))`
`=1-2cosC.cos(A-B)-2cos^(2)C`
`=1-2cosC[cos(A-B)+cosC]`
`=1-2cosC.2sinAsinB`
`=1-4sinAsinBsinC` Hence Proved.
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