In a ABC- if cos Acos Bcos C -3 - 18 and sin Asin Bsin C - 3-38- then The value of tan Atan B-tan B tan C-tan Ctan A is

Cos (A+B+C) nbsp;= cos Acos Bcos C[1 tan Atan B tan Btan C tan Ctan A] ⇒ 1=√(3) 18× nbsp; [1 tan Atan B tan Btan C tan Ctan A] ⇒ nbsp;1 tan Atan B ta ...

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Standard XI

Mathematics

Conditional Identities

In a ABC, if ...

Question

In a $△ A B C$ , if $cos A cos B cos C = \frac{\sqrt{3} - 1}{8}$ and $sin A sin B sin C = \frac{3 + \sqrt{3}}{8}$ , then

The value of $tan A tan B + tan B tan C + tan C tan A$ is

5 - 4 \sqrt{3}

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5 + 4 \sqrt{3}

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6 + \sqrt{3}

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6 - \sqrt{3}

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Solution

The correct option is B $5 + 4 \sqrt{3}$
$cos (A + B + C) = cos A cos B cos C [1 - tan A tan B - tan B tan C - tan C tan A] \Rightarrow - 1 = \frac{\sqrt{3} - 1}{8} \times [1 - tan A tan B - tan B tan C - tan C tan A] \Rightarrow 1 - tan A tan B - tan B tan C - tan C tan A = \frac{8}{1 - \sqrt{3}} \Rightarrow tan A tan B + tan B tan C + tan C tan A = 1 - \frac{8}{1 - \sqrt{3}} = \frac{- 7 - \sqrt{3}}{1 - \sqrt{3}} = \frac{7 + \sqrt{3}}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = 5 + 4 \sqrt{3}$

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In a △ABC, if cosAcosBcosC=√3−18 and sinAsinBsinC=3+√38, then The value of tanAtanB+tanBtanC+tanCtanA is

The correct option is B 5+4√3cos(A+B+C)=cosAcosBcosC[1−tanAtanB−tanBtanC−tanCtanA]⇒−1=√3−18×[1−tanAtanB−tanBtanC−tanCtanA]⇒1−tanAtanB−tanBtanC−tanCtanA =81−√3⇒tanAtanB+tanBtanC+tanCtanA =1−81−√3=−7−√31−√3 =7+√3√3−1×√3+1√3+1=5+4√3

In a $△ A B C$ , if $cos A cos B cos C = \frac{\sqrt{3} - 1}{8}$ and $sin A sin B sin C = \frac{3 + \sqrt{3}}{8}$ , then

The value of $tan A tan B + tan B tan C + tan C tan A$ is