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Let $0 \le x \le \frac{1}{2}$. What is $$\tan \left( {{{\sin }^{ - 1}}\left( {\frac{x}{{\sqrt 2 }} + \sqrt {\frac{{1 - {x^2}}}{2}} } \right) - {{\sin }^{ - 1}}x} \right)$$?

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  • $\begingroup$ Um, plug your formula into a calculator? $\endgroup$ – hmakholm left over Monica Mar 8 '17 at 20:44
  • $\begingroup$ Clarification/effort/context would go a long way here $\endgroup$ – Brevan Ellefsen Mar 8 '17 at 20:44
  • $\begingroup$ Are you looking for a series here? I am fairly confident this does not simplify into a nicer looking closed form $\endgroup$ – Brevan Ellefsen Mar 8 '17 at 20:46
  • $\begingroup$ @HenningMakholm - No calculator. $\endgroup$ – Tp-link Mar 8 '17 at 20:46
  • $\begingroup$ @Tp-link Take the first few terms of the necessary Taylor Series then and all should be well? $\endgroup$ – Brevan Ellefsen Mar 8 '17 at 20:47
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Write $x$ as $\sin\theta$ with $0\le \theta\le \pi/6$. Then your expression equals

$$\tan \left( {{\sin }^{ - 1}}\left( \frac{\sin \theta}{{\sqrt 2 }} + \frac{\cos\theta}{\sqrt 2} \right) - \theta \right) $$ But $\sin \theta + \cos\theta = \sqrt2 \sin(\theta+\pi/4)$ (see here), so the remaining arcsine is just $\sin^{-1}(\sin(\theta+\pi/4))$ and what you're left with is $$ \tan( \theta+\pi/4 - \theta ) = \tan(\pi/4) = 1 $$ no matter what $x$ is, as long as $\theta+\pi/4 \le \pi/2$.

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Henning's way is fantastic and ought to be chosen as the correct answer.
Nevertheless, here is another, more naive route.


We start by noting that, for all $x$ in your interval, $$\arcsin\left(\frac{\sqrt{1-x^2}+x}{\sqrt 2} \right) = \arcsin (x) + \pi/4$$ To prove this we first rearrange and note that $\arcsin$ is an odd function, meaning that we wish to prove $$\arcsin\left(\frac{\sqrt{1-x^2}+x}{\sqrt 2} \right)+\arcsin (-x)=\pi/4$$ On the LHS we use the Sum of Arcsines identity to rewrite the LHS as $$\arcsin\left(\frac{\sqrt{1-x^2}+x}{\sqrt 2} \right)+\arcsin (-x)\\= \arcsin \left(\frac{\sqrt{1-x^2}+x}{\sqrt 2}\sqrt{1-x^2}-x\sqrt{1-\left(\frac{\sqrt{1-x^2}+x}{\sqrt 2}\right)^2}\right)$$ With a little bit of work, we can show that the inside evaluates to $\sqrt{2}/2$, on the interval $[-\sqrt{2}/2,\sqrt{2}/2]$. We thus get $\tan(\arcsin(\sqrt{2}/2))=\tan(\pi/4)=1$

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