Principle of Mathematical Induction

Principle of Mathematical Induction

Question

In any triangle ABC, prove that

left parenthesis straight a plus straight b plus straight c right parenthesis space open square brackets tan open parentheses straight A over 2 close parentheses plus tan open parentheses straight B over 2 close parentheses close square brackets space equals space 2 straight c space cot open parentheses straight C over 2 close parentheses


Answer
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 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 a plus straight b plus straight c right parenthesis space open square brackets tan open parentheses straight A over 2 close parentheses plus tan open parentheses straight B over 2 close parentheses close square brackets space equals space 2 straight s open square brackets square root of fraction numerator left parenthesis straight s minus straight b right parenthesis 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 plus 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 end root close square brackets
                                                                                                   (∵ a + b + c = 2s)
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                         = 2 square root of straight s square root of straight s minus straight c end root open square brackets fraction numerator 2 straight s minus straight a minus straight b over denominator square root of left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis end root end fraction close square brackets space equals space 2 square root of straight s space end root space square root of straight s minus straight c end root open square brackets fraction numerator straight a plus straight b plus straight c minus straight a minus straight b over denominator square root of left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis end root end fraction close square brackets
                         = space space space 2 straight c square root of fraction numerator straight s left parenthesis straight s minus straight c right parenthesis over denominator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis end fraction end root space equals space 2 straight c space cot open parentheses straight C over 2 close parentheses  open square brackets because space 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 equals space tan open parentheses straight C over 2 close parentheses close square brackets
Hence,    left parenthesis straight a plus straight b plus straight c right parenthesis space open square brackets tan open parentheses straight A over 2 close parentheses plus tan open parentheses straight B over 2 close parentheses close square brackets space equals space 2 straight c space cot open parentheses straight C over 2 close parentheses

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