The value of tan 5x tan 3x tan 2x – tan 5x + tan 3x + tan 2x is ____________.

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Solution

$Consider, \tan 5 x \tan 3 x \tan 2 x - \tan 5 x + \tan 3 x + \tan 2 x . . . (1) Since 5 x = 3 x + 2 x i . e . \tan 5 x = \tan (3 x + 2 x) \tan 5 x = \frac{\tan 3 x + \tan 2 x}{1 - \tan 3 x \tan 2 x} i . e . \tan 5 x (1 - \tan 3 x \tan 2 x) = \tan 3 x + \tan 2 x i . e . \tan 5 x - \tan 5 x \tan 3 x \tan 2 x = \tan 3 x + \tan 2 x i . e . \tan 5 x \tan 3 x \tan 2 x - \tan 5 x + \tan 3 x + \tan 2 x = 0 Hence value of \tan 5 x \tan 3 x \tan 2 x - \tan 5 x + \tan 3 x + \tan 2 x is 0 .$

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