Principle Of Mathematical Induction

Question
CBSEENMA11014343

In any triangle ABC, prove that

cos space straight A space equals space fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction comma space cos space straight B space equals space fraction numerator straight c squared plus straight a squared minus straight b squared over denominator 2 ac end fraction space and space cos space straight C space equals space fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction

where a, b and c are as usual the sides of the triangle.

Solution

Proof:
          
Let angle C be acute in figure (i), obtuse in figure (ii) and right angle in figure
(iii)  Draw AD perpendicular BC in all the triangles (on produced if necessary)
                          straight c squared space equals space AB squared space equals space BD squared plus AD squared                                                      ....(i)
In figure (i),      space space AD over AC equals sinC
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In figure (iii),        AD over AC space equals space 1 space equals space sin 90 degree space equals space sinC
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∴ In each figure, AD = b sin C                                                                         ....(ii)
In figure (ii),         BD space equals space BC minus DC space equals space BC minus AC space. DC over AC space equals space straight a minus straight b space cosC
In figure (i),        space space BD space equals space BC plus DC space equals space BC plus AC. DC over AC equals straight a plus straight b space cos left parenthesis straight pi minus straight C right parenthesis space equals space straight a minus straight b space cosC
In figure (iii),         BD space equals space BC space equals space straight a minus straight b space cos 90 degree space equals space straight a minus straight b space cosC
∴ In each figure,    BD space equals space straight a minus straight b space cosC                                                              ....(iii)
Using (ii) and (iii) in (i), we get
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rightwards double arrow                   straight c squared equals straight a squared plus straight b squared cos squared straight C minus 2 ab space cosC plus straight b squared sin squared straight C
rightwards double arrow             2 ab space cosC space equals space straight a squared plus straight b squared left parenthesis cos squared straight C plus sin squared straight C right parenthesis minus straight c squared
rightwards double arrow             2 ab space cosC space equals space straight a squared plus straight b squared minus straight c squared
rightwards double arrow                     cosC space equals space fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction
Similarly,          cosA equals fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction space space and space space cosB space equals space fraction numerator straight a squared plus straight c squared minus straight b squared over denominator 2 ac end fraction
Hence proved.

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