Find all real numbers $x$ such that $$\sin(\arccos(\tan(\arcsin x))))=x.$$

Using [Wolfram|Alpha](http://www.wolframalpha.com/input/?i=sin(arccos(tan(arcsinx)))) we get $$\sin(\arccos(\tan(\arcsin x))))=\sqrt{\dfrac{2x^2-1}{x^2-1}}$$

Why? Because some case such $\arcsin{x}=\arccos{(1-x^2)}$ or $\pi-\arccos{(\sqrt{1-x^2})}.$

I think the answer is $x=\dfrac{\sqrt{5}\pm 1}{2}.$ Am I right?