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If A+B+C = pi, prove that : cosA+cosB + ...

If A+B+C = `pi`, prove that : `cosA+cosB + cosC = 1+4sinA/2sinB/2sinC/2`.

Text Solution

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LHS`=cosA+cosB+cosC`
`=2cos(A+B)/(2)cos(A-B)/(2)+(1-2sin^(2)C/2)`
`=1+2cos(180^(@)-C)/(2)cos(A-B)/(2)-2sinC/2.sin(180^(@))/(2)` `(therefore A+B+C=180^(@))`
`=1+2cos(90^(@)-C/2)cos(A-B)(2)-2sinC/2sin(90^(@)-(A+B)/(2))`
`=1+2sinC/2.2sinA/2.sinB/2`
`=1+4sinA/2sinB/2sinC/2` =RHS Hence Proved.
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