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

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:

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

Solution 1:

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 $c^2=1-x^2$

Solution 2:

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 $\sf{x}$.

Solution 3:

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 $$\left\{\left\{x\to -\sqrt{-\frac{7}{4}+\frac{3}{\sqrt{2}}-\frac{1}{4} \sqrt{97-68 \sqrt{2}}}\right\},\left\{x\to \sqrt{-\frac{7}{4}+\frac{3}{\sqrt{2}}-\frac{1}{4} \sqrt{97-68 \sqrt{2}}}\right\}\right\}$$

Solution 4:

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