In A B C prove that A B+ B + C A=1 - Vidyadip

Question

In $\triangle A B C$ prove that
$\cot A \cot B+\cot B \cot +\cot C \cot A=1$

✓

Answer

In $\triangle A B C, \mathrm{~A}+\mathrm{B}+\mathrm{C}=\pi$
$
\begin{aligned}
& \therefore \mathrm{A}+\mathrm{B}=\pi-\mathrm{C} \\
& \therefore \tan (A+B)=\tan (\pi-\mathrm{C}) \\
& \therefore \frac{\tan A+\tan B}{1-\tan A \tan B}=\tan (\pi-\mathrm{C}) \\
& \therefore \tan A+\tan B=-\tan C+\tan A \tan B \tan C \\
& \therefore \tan A+\tan B+\tan C=\tan A \tan B \tan C \\
& \therefore \frac{1}{\cot A}+\frac{1}{\cot B}+\frac{1}{\cot C}=\frac{1}{\cot A} \cdot \frac{1}{\cot B} \cdot \frac{1}{\cot C} \\
& \therefore \cot A \cot B+\cot B \cot C+\cot C \cot A=1
\end{aligned}
$

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