If A+B+C = 180 degree , then what is the value of Sigma CotA + CotB / TanA + TanB =?

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Solution

$Dear Student, \sum \frac{cotA + cotB}{tanA + tanB} = \sum \frac{cotA + cotB}{\frac{1}{cotA} + \frac{1}{cotB}} = \sum \frac{cotA + cotB}{cotA + cotB} \times cotAcotB = \sum cotAcotB = cotAcotB + cotBcotC + cotCcotA$
Now,
given A+B+C = 180°

implies A + B = 180°- C

implies tan(A + B) = tan(180°- C)

using trigonometry formulae

tan(A + B) = -tan(C)

therefore

[tanA +tanB] / [1-tanAtanB ] = -tanC

cross multiplying

tanA +tanB = -tanC[1-tanAtanB ]

tanA +tanB = -tanC +tanAtanBtanC

or

tanA +tanB + tanC = tanAtanBtanC

divide each term with tanAtanBtanC we get

cot A cotB + cotBcotC +cotC cotA=1

Regards

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