Principle of Mathematical Induction

Principle of Mathematical Induction

Question

In any triangle ABC, prove that

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Answer
space space space tan open parentheses straight A over 2 close parentheses space equals space square root of fraction numerator left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis over denominator straight s left parenthesis straight s minus straight a right parenthesis end fraction end root comma space space space space space space space space space space space tan open parentheses straight B over 2 close parentheses space equals space square root of fraction numerator left parenthesis straight s minus straight c right parenthesis space left parenthesis straight s minus straight a right parenthesis over denominator straight s left parenthesis straight s minus straight b right parenthesis end fraction end root
   tan open parentheses straight C over 2 close parentheses space equals space square root of fraction numerator left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis over denominator straight s left parenthesis straight s minus straight c right parenthesis end fraction end root

left parenthesis straight s minus straight a right parenthesis space tan open parentheses straight A over 2 close parentheses space equals space left parenthesis straight s minus straight a right parenthesis square root of fraction numerator left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis over denominator straight s space left parenthesis straight s minus straight a right parenthesis end fraction end root space equals space square root of fraction numerator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis over denominator straight s end fraction end root                       ...(i)
left parenthesis straight s minus straight b right parenthesis space tan open parentheses straight B over 2 close parentheses space equals space left parenthesis straight s minus straight b right parenthesis space square root of fraction numerator left parenthesis straight s minus straight c right parenthesis left parenthesis straight s minus straight a right parenthesis over denominator straight s left parenthesis straight s minus straight b right parenthesis end fraction space end root space equals space square root of fraction numerator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis over denominator straight s end fraction end root                       ...(ii)
left parenthesis straight s minus straight c right parenthesis space tan open parentheses straight C over 2 close parentheses space equals space left parenthesis straight s minus straight c right parenthesis square root of fraction numerator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis over denominator straight s left parenthesis straight s minus straight c right parenthesis end fraction end root space equals space square root of fraction numerator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis over denominator straight s end fraction end root                         ...(iii)
Hence, from (i) and (ii), we get
      left parenthesis straight s minus straight a right parenthesis space tan open parentheses straight A over 2 close parentheses space equals space left parenthesis straight s minus straight b right parenthesis space tan open parentheses straight B over 2 close parentheses space equals space left parenthesis straight s minus straight c right parenthesis space tan open parentheses straight C over 2 close parentheses
  

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