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Solve $$\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $$



My attempt:

Let $A = \sqrt{1+x}, B = \sqrt{1-x}$ and then by squaring the problematic equation we get:

$$(1+AB)(A^{3} + B^{3})^{2} = (AB)^{2} + 4AB + 4 $$ $$ A^{6} + B^{6} + BA^{7} + A B^{7} = -2 (AB)^{4} - 2(AB)^{3} + (AB)^{2} + 4AB + 4 $$

I also have tried using$A = (1+x), B = (1-x)$ , and some others, but none solves the problem.



I am now trying $A = (1+x)$ and $(1-x) = -(1+x) + 2 = 2 - A$, so:

$$\sqrt{1 + \sqrt{A(2-A)}}\left(\sqrt{(A)^{3}} + \sqrt{(2-A)^{3}} \right) = 2 + \sqrt{A(2-A)} $$

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    $\begingroup$ First try may progress if you notice that $A^2+B^2=2$. $\endgroup$ – Nicolas FRANCOIS May 25 at 9:47
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    $\begingroup$ Where is this problem from? $\endgroup$ – Toby Mak May 25 at 9:49
  • $\begingroup$ Shouldn't it be $A^3 + B^3 $ in the line $(1+AB)(A^{3} - B^{3})^{2} = (AB)^{2} + 4AB + 4$ $\endgroup$ – Ishan Deo May 25 at 9:52
  • $\begingroup$ @IshanDeo thanks, right $\endgroup$ – Arief Anbiya May 25 at 9:55
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    $\begingroup$ Note: if you square the equation you need to be careful as you will likely get some additional solutions which will not be solutions to the original. This is because both $X=Y$ and $X=-Y$ square to $X^2=Y^2$ $\endgroup$ – Mark Bennet May 25 at 10:12
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First, this is stated as an equation to solve (for $x$) rather than an identity to be shown.

So with $a=\sqrt {1+x}$ and $b=\sqrt {1-x}$ we have $$a^2+b^2=2$$ and $$(a+b)^2=a^2+2ab+b^2=2(1+ab)$$ and $$a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(2-ab)$$

Then $$\sqrt {1+ab}\cdot (a^3+b^3)=\frac {\sqrt 2}2(a+b)(a+b)(2-ab)=\sqrt 2(1+ab)(2-ab)$$

If we then put $c=ab$ the equation to solve is then $$\sqrt 2(1+c)(2-c)=2+c$$ which is a straightforward quadratic in $c$. Then solve for $x$ by noting $c^2=1-x^2$

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    $\begingroup$ It would be $2+c$, not $1+c$ as $a^2+b^2=2$. $\endgroup$ – Vineet Mangal May 25 at 10:16
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Great substitution technique. Notice that $\sf{A=\sqrt{1+x}}$ and $\sf{B=\sqrt{1-x}}$ leads to $$\sf{(A^3+B^3)\sqrt{1+AB}}=2+AB$$ and since $\sf{A^3+B^3=(A+B)(A^2-AB+B^2)}$ and $\sf{A^2+B^2=2\implies (A+B)^2=2(1+AB)}$, we get $$\sf{(A+B)\sqrt{1+AB}=\frac{2+AB}{2-AB}}\implies (1+AB)\sqrt2=1+\frac{2AB}{2-AB}$$ which can be solved for $\sf{AB}$, and thus $\sf{x}$.

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Substituting $$a=\sqrt{1+x},b=\sqrt{1-x}$$ and using that $$a^2+b^2=2$$ and $$a^3+b^3=(a+b)(a^2+b^2-ab)$$ we get$$\sqrt{1+ab}(a+b)(2-ab)=2+ab$$ And we get by squaring $$(1+ab)(2+2ab)(2-ab)^2=(2+ab)^2$$ and with $$u=ab$$ we get $$2(1+u)^3(2-u)^2=(2+u)^2$$ The solutions are $$\left\{\left\{x\to -\sqrt{-\frac{7}{4}+\frac{3}{\sqrt{2}}-\frac{1}{4} \sqrt{97-68 \sqrt{2}}}\right\},\left\{x\to \sqrt{-\frac{7}{4}+\frac{3}{\sqrt{2}}-\frac{1}{4} \sqrt{97-68 \sqrt{2}}}\right\}\right\}$$

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  • $\begingroup$ I think the RHS should be $2 + ab$ $\endgroup$ – Arief Anbiya May 25 at 9:58
  • $\begingroup$ Sorry this was a typo. $\endgroup$ – Dr. Sonnhard Graubner May 25 at 9:59
  • $\begingroup$ But $$a^3+b^3=(a+b)(a^2+b^2-ab)=(a+b)(2-ab)$$ $\endgroup$ – Dr. Sonnhard Graubner May 25 at 10:08
  • $\begingroup$ Where is the factor in your line? $\endgroup$ – Dr. Sonnhard Graubner May 25 at 10:09
  • $\begingroup$ $$(a^3+b^3)=(a+b)(a^2+b^2-ab)=(a+b)(2-ab)$$ when we use that $$a^2+b^2=2$$ $\endgroup$ – Dr. Sonnhard Graubner May 25 at 10:11
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As $-1\le x\le1$

WLOG $x=\cos2t,0\le2t\le\pi,\sin2t=\sqrt{1-x^2}$

$$\implies\sqrt{1+\sin2t}[(2\cos^2t)^{3/2}+(2\sin^2t)^{3/2}]=2+\sin2t$$

As $\sin t,\cos t\ge0$ and $(\sin t+\cos t)^2=1+\sin2t$

$$2\sqrt2(\cos t+\sin t)(\cos^3t+\sin^3t)=2+\sin2t$$

$$\sqrt2(1+\sin2t)(2-\sin2t)=2+\sin2t$$ which is on rearrangement, a Quadratic Equation in $\sin2t(\ge0)$

as $\cos^3t+\sin^3t=(\cos t+\sin t)(\cos^2t+\sin^2t-\sin t\cos t)=\dfrac{(\cos t+\sin t)(2-\sin2t)}2$

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  • $\begingroup$ Nice point of view. $\endgroup$ – Arief Anbiya May 25 at 10:31
  • $\begingroup$ @trancelocation, As $0\le y\le\dfrac\pi2$ $$\sqrt{1+x}=\sqrt2\cos t$$ etc. $\endgroup$ – lab bhattacharjee May 25 at 14:24
  • $\begingroup$ @trancelocation, $$(\sqrt2\cos t)^3=2\sqrt2\cos^3t$$ etc. $\endgroup$ – lab bhattacharjee May 25 at 14:37
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Let $p = A+B = \sqrt{1+x}+\sqrt{1-x}$. We have $$ p^2 = 2+2\sqrt{1-x^2}$$ $$ p^3 = \sqrt{(1-x)^3}+\sqrt{(1+x)^3} + 3p\sqrt{1-x^2}$$ so $$ 1+ \sqrt{1-x^2} = \frac12 p^2$$ $$ \sqrt{(1-x)^3}+\sqrt{(1+x)^3} = -\frac12 p^3 +3p$$ and we get $$ \frac{p}{\sqrt{2}} (-\frac12 p^3 +3p) = 1 + \frac12 p^2$$ $$ - \frac{1}{2\sqrt{2}}p^4 +\frac{3\sqrt{2}-1}{2} p^2 - 1 = 0$$ which can be solved for $p$.

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Note: $1-x^2\ge 0 \Rightarrow -1\le x\le 1$.

Square both sides and denote $1-x^2=t^2, -1\le t\le 1$: $$\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} \Rightarrow \\ (1+\sqrt{1-x^2})(2+6x^2+2\sqrt{(1-x^2)^3}=5-x^2+4\sqrt{1-x^2} \Rightarrow \\ (1+t)(2+6(1-t^2)+2t^3)=4+t^2+4t \Rightarrow \\ 2t^4-4t^3-7t^2+4t+4=0 \Rightarrow \\ t\approx 0.93, x\approx \pm 0.38.$$ Note: Other roots are rejected as they are outside of domain.

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  • $\begingroup$ Something must have gone wrong since $0.77$ does not seem to be close to a solution: wolframalpha.com/input/… $\endgroup$ – JustAnotherStackUser May 25 at 10:27
  • $\begingroup$ Right, $\pm 0.77$ don't fit, so they are gone now. $\endgroup$ – farruhota May 25 at 10:31
  • $\begingroup$ $+0.38$ should also be a solution $\endgroup$ – JustAnotherStackUser May 25 at 10:32
  • $\begingroup$ Right again, since it is within the doman $x\in [-1,1]$. Thank you $\endgroup$ – farruhota May 25 at 10:34
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Let us first simplify the expression. Assume $a^2=1-x$ $ $ & $ $ $b^2=1+x$

Please note that $a^2+b^2=2$

Now our equation becomes $\sqrt{1+ab}(a^3+b^3)=2+ab$

$\sqrt{\frac{a^2+b^2+2ab}{2}}(a+b)(a^2+b^2-ab)=2+ab$ which further simplifies to

$\frac{(a+b)^2}{\sqrt{2}}(2-ab)=2+ab$

$\frac{(2+2ab)(2-ab)}{\sqrt{2}}=2+ab$

Find $ab$ and use the equation $a^2b^2=1-x^2$

Hope this will be helpful!

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    $\begingroup$ $(a+b)^2=2+2ab$, not $2+ab$. $\endgroup$ – mathlove May 25 at 10:22
  • $\begingroup$ Typo Corrected! $\endgroup$ – Vineet Mangal May 25 at 10:40

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