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