Find the Value of tan 5x tan 3x tan 2x − tan 5x + tan 3x + tan 2x

Solution

Using the identity:

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

Take

\[ A=3x, \qquad B=2x \]

Then,

\[ \tan5x = \frac{ \tan3x+\tan2x } { 1-\tan3x\tan2x } \]

Cross multiplying,

\[ \tan5x(1-\tan3x\tan2x) = \tan3x+\tan2x \]

Expanding,

\[ \tan5x – \tan5x\tan3x\tan2x = -\tan3x-\tan2x \]

Rearranging,

\[ \tan5x\tan3x\tan2x -\tan5x +\tan3x +\tan2x =0 \]

Therefore,

\[ \boxed{0} \]