If $A + B + C = 180^{o}$ , and $tan A + tan B + tan C$ = $k tan A tan B tan C$ , then value of $k$ is

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Solution

Given that $A + B + C = 180$

$⟹ A + B = 180 - C$

Applying tan on both sides

$⟹ t a n (A + B) = t a n (180 - C)$

$⟹ \frac{t a n A + t a n B}{1 - t a n A t a n B} = - t a n C$

$⟹ t a n A + t a n B = - t a n C (1 - t a n A t a n B)$

$⟹ t a n A + t a n B = - t a n C + t a n A t a n B t a n C$

$⟹ t a n A + t a n B + t a n C = t a n A t a n B t a n C$

$⟹ k = 1$

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