Find x in the irrational equation below: $$\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$$ Answer: $x = \cos 54^\circ$
I try: $$\begin{align} \sqrt{1-x} &=2x^2-1+2x\sqrt{1-x^2} \tag1 \\ \sqrt{1-x} &=2x^2-1+2x\sqrt{(1-x)(1+x)} \tag2 \\ \sqrt{1-x}-2x\sqrt{(1-x)(1+x)} &=2x^2-1 \tag3 \\ 1-x-4x(\sqrt{1-x}\sqrt{(1-x)(1+x)}+4x^2(1-x^2)&=4x^4-4x^2+1 \tag4 \\ 4x^4-4x^2+4x(1-x)\sqrt{(1+x)}-4x^2+4x^4 &=0 \tag5 \\ 8x^4-8x^2+4x(1-x)\sqrt{(1+x)} &=0 \tag6 \\ 2x^4-2x^2+(1-x)\sqrt{(1-x)} &=0 \tag7 \\ &\text{???} \end{align}$$
