Principle of Mathematical Induction

Principle of Mathematical Induction

Question

In any triangle ABC, show that

cosA over straight a plus cosB over straight b plus cosC over straight c space equals space fraction numerator straight a squared plus straight b squared plus straight c squared over denominator 2 abc end fraction

Answer

In any ΔABC comma space
                cosA space equals space fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction
rightwards double arrow            cosA over straight a space equals space fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 abc end fraction                                                   ...(i)
Similarly,      fraction numerator cos space straight B over denominator straight b end fraction space equals space fraction numerator straight c squared plus straight a squared minus straight b squared over denominator 2 abc end fraction                                                 ...(ii)
and               fraction numerator cos space straight C over denominator straight c end fraction space equals space fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 abc end fraction                                               ...(iii)
Adding (i), (ii) and (iii), we get
        L.H.S = space space cosA over straight a plus cosB over straight b plus cosC over straight c
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                equals fraction numerator straight a squared plus straight b squared plus straight c squared over denominator 2 abc end fraction space equals space straight R. straight H. straight S.
Hence,       cosA over straight a plus cosB over straight b plus cosC over straight c space equals space fraction numerator straight a squared plus straight b squared plus straight c squared over denominator 2 abc end fraction

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