Given that (1+(√(1+x)))tan x=(1+(√(1−x))). Then find sin 4x.


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Question Number 62109 by sandhyavs last updated on 15/Jun/19

$G i v e n t h a t$ $(1 + \sqrt{1 + x}) \tan x = (1 + \sqrt{1 - x}) .$ $T h e n f i n d \sin 4 x .$
Commented by sandhyavs last updated on 15/Jun/19

$A l s o t e l l m e t h e s t e p s . P l e a s e . . .$
Commented by sandhyavs last updated on 15/Jun/19

$p l e a s e a n s w e r m e . . .$
	Answered by MJS last updated on 15/Jun/19

	$\tan x = t$ $(1 + \sqrt{1 + x}) t = 1 + \sqrt{1 - x}$ $\sqrt{1 - x} - t \sqrt{1 + x} = t - 1$ $squaring and transforming$ $2 t \sqrt{1 - x} \sqrt{1 + x} = (t^{2} - 1) x + 2 t$ $squaring and transforming$ $({(t^{2} + 1)}^{2} x + 4 t (t^{2} - 1)) x = 0$ $\Rightarrow x = 0 \lor x = \frac{4 t (1 - t) (1 + t)}{{(t^{2} + 1)}^{2}}$ $t = \tan x$ $x = 4 \sin x \cos x (\cos^{2} x - \sin^{2} x) = \sin 4 x$
	Commented by sandhyavs last updated on 15/Jun/19

	$t h a n k y o u s o m u c h .$
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