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Question
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
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Solution
We know that 8x = 6x + 2x
Therefore,
\[ \tan\left( 8x \right) = \tan\left( 6x + 2x \right)\]
\[ \Rightarrow \tan\left( 8x \right) = \frac{\tan6x + \tan2x}{1 - \tan6x \tan2x}\]
\[ \Rightarrow \tan8x - \tan8x \tan6x \tan2x = \tan6x + \tan2x\]
\[ \Rightarrow \tan8x - \tan6x - \tan2x = \tan8x \tan6x \tan2x\]
Hence proved.
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