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Prove that tan (2 tan^(-1) x ) = 2 tan...

Prove that ` tan (2 tan^(-1) x ) = 2 tan (tan^(-1) x + tan^(-1) x^(3))`.

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To prove that \( \tan(2 \tan^{-1} x) = 2 \tan(\tan^{-1} x + \tan^{-1} x^3) \), we will start by simplifying both sides of the equation step by step. ### Step 1: Simplify the Left-Hand Side We start with the left-hand side: \[ LHS = \tan(2 \tan^{-1} x) \] Using the double angle formula for tangent, we have: ...
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