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Sine Rule

Evaluate tan ...

Question

Evaluate $\tan 3 A - \tan 2 A - \tan A$

A

$\tan 3 A \tan 2 A \tan A$

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B

$- \tan 3 A \tan 2 A \tan A$

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C

$\tan A \tan 2 A - \tan 2 A \tan 3 A - \tan 3 A \tan A$

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D

None of the above

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Solution

The correct option is A
$\tan 3 A \tan 2 A \tan A$

Explanation for correct option
Evaluating the given trigonometric expression
Given, $\tan 3 A - \tan 2 A - \tan A$
We know that,
$3 A = 2 A + A$
Taking tan on both sides:
$\tan 3 A = \tan (2 A + A) \Rightarrow \tan 3 A = \frac{\tan 2 A + \tan A}{1 - \tan 2 A \tan A} [∵ t a n (A + B) = \frac{t a n A + t a n B}{1 - t a n A t a n B}] \Rightarrow t a n 3 A (1 - t a n 2 A t a n A) = t a n 2 A + t a n A$
Cross multiplying and simplifying further, we get,
$\Rightarrow \tan 3 A - \tan 3 A \tan 2 A \tan A = \tan 2 A + \tan A$
$∴ \tan 3 A - \tan 2 A - tanA = \tan 3 A \tan 2 A tanA$
Hence, the correct answer is Option (A).

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