How do you simplify the expression : arcsin((x+1)/sqrt(2*(x²+1))) ?

I tried many ways but i just can't find the right way to do it, so just give me a hand please :)

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dk_ch Share
Oct 29, 2016

Given expression =arcsin((x+1)/sqrt(2*(x²+1)))
Let $x = \tan \theta$

So $\theta = {\tan}^{-} 1 x$

Inserting $x = \tan \theta$ the given expression becomes

$= \arcsin \left(\frac{\tan \theta + 1}{\sqrt{2 \cdot \left({\tan}^{2} \theta + 1\right)}}\right)$

$= \arcsin \left(\frac{\sin \frac{\theta}{\cos} \theta + 1}{\sqrt{2 \cdot \left({\sec}^{2} \theta\right)}}\right)$

$= \arcsin \left(\frac{1}{\sqrt{2}} \left(\frac{\sin \theta \sec \theta + 1}{\sec \theta}\right)\right)$

$= \arcsin \left(\frac{1}{\sqrt{2}} \left(\frac{\sin \theta \cancel{\sec} \theta}{\cancel{\sec}} \theta + \frac{1}{\sec \theta}\right)\right)$

$= \arcsin \left(\frac{1}{\sqrt{2}} \left(\sin \theta + \cos \theta\right)\right)$

$= \arcsin \left(\frac{1}{\sqrt{2}} \sin \theta + \frac{1}{\sqrt{2}} \cos \theta\right)$

$= \arcsin \left(\cos \left(\frac{\pi}{4}\right) \sin \theta + \sin \left(\frac{\pi}{4}\right) \cos \theta\right)$

$= \arcsin \left(\sin \left(\theta + \frac{\pi}{4}\right)\right)$

$= \theta + \frac{\pi}{4}$

$= {\tan}^{-} 1 x + \frac{\pi}{4}$

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