In triangle ABC- aA-bB-cC-

a A+b B+c C =2Rsin A A+2Rsin B B+2Rsin C C =2Rcos A+2Rcos B+2Rcos C =2R(cos A+cos B+cos C) =2R(1+4sinA2sinB2sinC2) =2R(1+rR) ...

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Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

In triangle A...

Question

In triangle ABC, $a cot A + b cot B + c cot C =$

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Solution

$a cot A + b cot B + c cot C$
$= 2 R sin A cot A + 2 R sin B cot B + 2 R sin C cot C$
$= 2 R cos A + 2 R cos B + 2 R cos C$
$= 2 R (cos A + cos B + cos C)$
$= 2 R (1 + 4 sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2})$
$= 2 R (1 + \frac{r}{R})$
$= 2 (R + r)$

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In triangle ABC, acotA+bcotB+ccotC=

acotA+bcotB+ccotC=2RsinAcotA+2RsinBcotB+2RsinCcotC=2RcosA+2RcosB+2RcosC=2R(cosA+cosB+cosC)=2R(1+4sinA2sinB2sinC2)=2R(1+rR)=2(R+r)

In triangle ABC, $a cot A + b cot B + c cot C =$

$a cot A + b cot B + c cot C$
$= 2 R sin A cot A + 2 R sin B cot B + 2 R sin C cot C$
$= 2 R cos A + 2 R cos B + 2 R cos C$
$= 2 R (cos A + cos B + cos C)$
$= 2 R (1 + 4 sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2})$
$= 2 R (1 + \frac{r}{R})$
$= 2 (R + r)$