show-that-tan-1-1-x-2-1-x-1-2-tan-1-x-

Question Number 1157 by navajyoti.tamuli.tamuli@gmail. last updated on 06/Jul/15

s h o w t h a t

{t a n}^{- 1} (\frac{\sqrt{1 + x^{2}} - 1}{x}) = \frac{1}{2} {t a n}^{- 1} x

Answered by prakash jain last updated on 11/Jul/15

Let y = \tan^{- 1} x \Rightarrow x = tany

1 + x^{2} = \sec^{2} y

\frac{\sqrt{1 + x^{2}} - 1}{x} = \frac{\sqrt{\sec^{2} y} - 1}{tany} = \frac{secy - 1}{tany} = \frac{1 - cosy}{siny}

= \frac{1 - (1 - 2 \sin^{2} \frac{y}{2})}{2 \sin \frac{y}{2} \cos \frac{y}{2}} = \tan \frac{y}{2}

L . H . S = \tan^{- 1} \tan \frac{y}{2} = \frac{y}{2}

R . H . S = \frac{1}{2} \tan^{- 1} x = \frac{y}{2}

∴ L . H . S = R . H . S