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

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 .