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Home / Doubts and Solutions/ICSE/Class 11 Science/Mathematics

prove a square minus b square divided by cos a + cos b + b square minus c square divided by cos b + cos c + c square minus a square divided by cos c + cos a = to 0

Asked by adityabhardwaj8239 14th June 2020, 7:22 PM
Answered by Expert
Answer:
(a2-b2)/(cosA+cosB) + (b2-c2)/(cosB+cosC) + (c2-a2)/(cosA+cosC)
According to the sine rule,
a/sinA = b/sinB = c/sinC = k
a = k sinA, b = k sinB and c = k sinCfraction numerator a squared minus b squared over denominator cos A plus cos B end fraction plus fraction numerator b squared minus c squared over denominator cos B plus cos C end fraction plus fraction numerator c squared minus a squared over denominator cos C plus cos A end fraction equals fraction numerator k squared space sin squared A minus k squared space sin squared B over denominator cos A plus cos B end fraction plus fraction numerator k squared space sin squared B minus k squared space sin squared C over denominator cos B plus cos C end fraction plus fraction numerator k squared space sin squared C minus k squared space sin squared A over denominator cos C plus cos A end fraction equals fraction numerator k squared space open parentheses 1 minus cos squared A minus 1 plus cos squared B close parentheses over denominator cos A plus cos B end fraction plus fraction numerator k squared open parentheses 1 minus cos squared B minus 1 plus cos squared C close parentheses over denominator cos B plus cos C end fraction plus fraction numerator k squared space open parentheses 1 minus cos squared C minus 1 plus cos squared A close parentheses over denominator cos C plus cos A end fraction equals k squared open square brackets fraction numerator open parentheses cos squared B minus cos squared A close parentheses over denominator cos A plus cos B end fraction plus fraction numerator open parentheses cos squared C minus cos squared B close parentheses over denominator cos B plus cos C end fraction plus fraction numerator open parentheses cos squared A minus cos squared C close parentheses over denominator cos C plus cos A end fraction close square brackets equals k squared open square brackets fraction numerator open parentheses cos B minus cos A close parentheses open parentheses cos A plus cos B close parentheses over denominator cos A plus cos B end fraction plus fraction numerator open parentheses cos C minus cos B close parentheses open parentheses cos C plus cos B close parentheses over denominator cos B plus cos C end fraction plus fraction numerator open parentheses cos A minus cos C close parentheses open parentheses cos A plus cos C close parentheses over denominator cos C plus cos A end fraction close square brackets equals k squared open square brackets cos B minus cos A plus cos C minus cos B plus cos A minus cos C close square brackets equals k squared open square brackets 0 close square brackets equals 0 H e n c e comma space fraction numerator a squared minus b squared over denominator cos A plus cos B end fraction plus fraction numerator b squared minus c squared over denominator cos B plus cos C end fraction plus fraction numerator c squared minus a squared over denominator cos C plus cos A end fraction equals 0
Answered by Expert 15th June 2020, 11:38 AM
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