Question

Prove that
2+tan-1x =tan-1 2x/1-x2
​

Answer 1

Answer:

$$\begin{lgathered}let \ \tan^{-1} x = \theta \\ \\ Then \ \ , \tan \theta = x \\ \\ We\ know\ that , \\ \\ \tan 2\theta =\bigl ( \frac{2 \tan \theta}{1- \tan^2 \theta} \bigr ) \\ \\ 2\theta = \tan^{-1} \bigl ( \frac{2\tan \theta}{1- \tan^2 \theta} \bigr ) \\ \\ Now, substitute \ the\ value\ of \ \tan \theta \ and \ \theta . \\ \\ Then, \\ \\ 2\tan^{-1}x = \tan^{-1} \bigl ( \frac{2x}{1-x^2} \bigr )\end{lgathered}$$

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

Putting x= tan(α) in RHS,

RHS=

$\tan ^{ - 1} ( \frac{2x}{1 - {x}^{2} } ) = \tan^{ - 1} ( \frac{2 \tan( \alpha ) }{1 - \tan^{2} ( \alpha ) } )$

$= \tan^{ - 1} ( \tan(2 \alpha ) )$

$= 2 \alpha = 2 \tan^{ - 1} ( x)$

as, x=tan(α)=>α=tan^(-1) (x)