Question

$\mathrm{In}\mathrm{triangle}\mathrm{ABC},\mathrm{if}\mathrm{cotA}.\mathrm{cotB}=\frac{1}{2}\mathrm{and}\mathrm{cotB}.\mathrm{cotC}=\frac{1}{18},\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{tanB}\mathrm{is}$

• A.
  3
• B.
  4
• C.
  5
• D.
  6

Answer 1

The correct option is B

4
Here, if we assume $\tan B=x$, then
$\tan A=\frac{2}{x}\mathrm{and}\tan C=\frac{18}{x}$.
We know the conditional identity for a triangle $ABC$.
$\tan A.\tan B.\tan C=\tan A+\tan B+\tan C.$
Now, substituting the values of $\frac{2}{x}+x+\frac{18}{x}=\frac{2}{x}.x.\frac{18}{x}\Rightarrow x+\frac{20}{x}=\frac{36}{x}\Rightarrow x=\frac{36}{x}-\frac{20}{x}\Rightarrow x=\frac{16}{x}\Rightarrow x=4=\tan B(\because cotA.cotB=1/2,so\tan B>0)$
Thus, Option b. is correct.