If A + B + C = π, then (tan A + tan B + tan C)/(tan A tan B tan C) is equal to (a) tan A tan B tan C (b) 0 (c) 1 (d) none of these

If A + B + C = π, Find (tan A + tan B + tan C)/(tan A tan B tan C)

Question:
If \[ A+B+C=\pi \] then \[ \frac{\tan A+\tan B+\tan C} {\tan A\tan B\tan C} \] is equal to

(a) \(\tan A\tan B\tan C\)
(b) \(0\)
(c) \(1\)
(d) none of these

Solution

Since

\[ A+B+C=\pi \]

therefore,

\[ \tan(A+B+C)=\tan\pi=0 \]

Using the tangent formula for three angles:

\[ \tan(A+B+C) = \frac{ \tan A+\tan B+\tan C -\tan A\tan B\tan C } { 1-\tan A\tan B-\tan B\tan C-\tan C\tan A } \]

Since the value is zero, numerator must be zero:

\[ \tan A+\tan B+\tan C -\tan A\tan B\tan C = 0 \]

Hence,

\[ \tan A+\tan B+\tan C = \tan A\tan B\tan C \]

Dividing both sides by

\[ \tan A\tan B\tan C \]

we get

\[ \frac{ \tan A+\tan B+\tan C } { \tan A\tan B\tan C } = 1 \]

\[ \boxed{ \frac{\tan A+\tan B+\tan C} {\tan A\tan B\tan C} =1 } \]

Correct Option: (c)

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