<s>[INST] simplification of trigonometric expression I search a simplification for 
$$ \arcsin\biggl( \frac{1}{\sqrt  2} \frac{x}{\sqrt  {(x-1)^2 +1}}\biggr)$$
any idea ? I tried composition, derivation and setting $x$ as a sine, cosine or a tangent in vain.
Thanks
 [/INST] Let us substitute  $x = 1 + \tan \theta $. We then get: $$E = \arcsin \left (\frac1 {\sqrt 2} \frac {1+\tan \theta}{\sqrt {\sec^2 \theta}} \right) $$ $$=\arcsin \left (\frac {1}{\sqrt 2} (\cos \theta + \sin \theta) \right) $$ $$=\arcsin \left (\sin \theta \cos \frac {\pi}{4} + \cos \theta \sin \frac {\pi}{4} \right) $$ $$=\arcsin \left (\sin (\theta + \frac {\pi}{4}) \right) $$ $$=\, ?$$
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