Cos(a+b)cosC-cos(b+c)cos a=
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cos(A + B).cosC - cos(B + C).cosA
= (cosAcosB - sinAsinB).cosC - (cosB.cosC - sinB.sinC).cosA
= cosA.cos.B.cosC - sinA.sin.B.sinC - cosA.cos.Bcos.C + sinB.sinC.cosA
= sinB.sinC.cosA - sinB.sinA.cosC
= sinB.(sinC.cosA - cosC.sinA)
= sinB.sin(C - A)
∴ cos(A + B).cosC - cos(B + C).cosA = sinB.sin(C - A)
Cheers!
cos(A + B).cosC - cos(B + C).cosA
= (cosAcosB - sinAsinB).cosC - (cosB.cosC - sinB.sinC).cosA
= cosA.cos.B.cosC - sinA.sin.B.sinC - cosA.cos.Bcos.C + sinB.sinC.cosA
= sinB.sinC.cosA - sinB.sinA.cosC
= sinB.(sinC.cosA - cosC.sinA)
= sinB.sin(C - A)
∴ cos(A + B).cosC - cos(B + C).cosA = sinB.sin(C - A)
Cheers!
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