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For any triangle ABC, prove that a(cosC-...

For any triangle ABC, prove that
`a(cosC-cos B)\ =\ 2\ (b- c)\ cos^2A/2`

Text Solution

Verified by Experts

`A(cosC-cosB)=2(b-c)cos^2(A/2)`
L.H.S`=a(cosC-cosB)`
`a/sinA=b/sinB=c/sinC=k`
`a=ksina,b=ksinB,C=ksinC`
Now L.H.S
`=k.2sin(A/2) cos(A/2)(2sin((C+B)/2))sin((B-C)/2))`
`=Kxx2sin(A/2)cos(A/2)xx2sin(pi-A)/2.sin((B-C)/2)`
`=4kxxsin(pi-((B+C))/2)xxcos(A/2)xxcos(A/2)xxsin((B-C)/2)`
`=4Kxxcos^2(A/2)xx(cos((B+C)/2)xxsin(B-C)/2)`
`=2Kxxcos^2(A/2)xx(sinB-sinC)`
`=2k(b/k-c/k)cos^2(A/2)`
`=2k((b-c)/k)cos^2(A/2)`
`=2(b-c)cos^2(A/2)`
=RHS
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