Question

Find $\cot A\cot B+\cot B\cot C+\cot C\cot A$.If $A+B+C=\pi$

• A. $1$
• B. $3$
• C. $4$
• D. None of these

Answer 1

The correct option is A $1$

$A+B+C=\pi$

$\Rightarrow A+B=\pi -C$

$\Rightarrow \tan \left(A+B\right)=\tan \left(\pi -C\right)$

$\Rightarrow \frac{\mathrm{tanA}+tanB}{1-\tan A\tan B}=-\tan C$

$\Rightarrow \mathrm{tanA}+tanB=-\tan C+\tan A\tan B\tan C$

$\Rightarrow \frac{\mathrm{tanA}+tanB+\tan C}{\tan A\tan B\tan C}=1$

$\Rightarrow \frac{\frac{1}{\cot A}+\frac{1}{\cot B}+\frac{1}{\cot C}}{\frac{1}{\cot A\cot B\cot C}}=1$

$\Rightarrow \cot A\cot B+\cot B\cot C+\cot C\mathrm{cotA}=1$