Show the $\arcsin$ identity: $ \arcsin(1 - 2x) + 2\arcsin(\sqrt{x}) = \pi / 2$

Question

Can somebody find an elementary proof of the following identity:

$$ \arcsin ( 1 - 2x) + 2 \arcsin(\sqrt{x}) = \frac\pi2 $$

I noticed it while solving the following integral:

$$ I = \int \frac{\sqrt{x}}{\sqrt{1 - x}} = -2 \sqrt{1 - x}\sqrt{x} + \int \frac{\sqrt{1 - x}}{\sqrt{x}} $$

where the first equality follows after applying an integration by parts with $f = \sqrt{x}$ and $\mathrm{d} g = 1/\sqrt{1 - x}$. For the sake of simplicity we omit $\mathrm{d}x$ in each integral. Adding the integral to itself:

\begin{align} 2I = \int \frac{\sqrt{x}}{\sqrt{1 - x}} &= -2 \sqrt{1 - x}\sqrt{x} + \int \left(\frac{\sqrt{1 - x}}{\sqrt{x}} + \frac{\sqrt{x}}{\sqrt{1- x}}\right) \\&= -2 \sqrt{1-x}\sqrt{x} + \int\frac{1}{\sqrt{x}\sqrt{1-x}}\end{align}

The last integral on the RHS evaluates to $\arcsin(1 - 2x)$, so $$I = - \sqrt{1-x}\sqrt{x} + \frac{1}{2}\arcsin(1 - 2x) + C.$$

On the other hand, the integral can also be evaluated by applying a $u$-sub with $u = \sqrt{x}$. We find that:

$$ I = 2 \int\frac{u^2}{\sqrt{1 - u^2}} =2 \left(-u \sqrt{1 - u^2} + \int\sqrt{1 - u^2}\right) = - u\sqrt{1 - u^2} + \arcsin u + C_2 $$

So then it follows that $I$ is also equal to $-\sqrt{x}\sqrt{1-x} + \arcsin{\sqrt{x}} + C_2$. Equate the results of the two methods and plug in a random point to find $C - C_2$ and the subsequent identity.

Answer 1

Here’s an elementary proof. Let $x=\sin^2\phi$. Then, since $\cos(2\phi)=1-2\sin^2\phi$, we have $$\arcsin(1-2x)=\arcsin(1-2\sin^2\phi)=\arcsin(\cos(2\phi))=\frac{\pi}{2}-2\phi$$ and $$2\arcsin(\sqrt{x})=2\arcsin(\sin\phi)=2\phi$$ From which the result follows elementarily (without calc).

Answer 2

Let $f(x)=\arcsin ( 1 - 2x) + 2 \arcsin\sqrt x$ where $x\in[0,1]$ so that \begin{align}f'(x)&=\frac1{\sqrt{1-(1-2x)^2}}\cdot(-2)+2\cdot\frac1{\sqrt{1-\sqrt x^2}}\cdot\frac1{2\sqrt x}\\&=-\frac2{\sqrt{4x-4x^2}}+\frac1{\sqrt{x(1-x)}}\\&=-\frac1{\sqrt{x(1-x)}}+\frac1{\sqrt{x(1-x)}}=0\end{align} for all $x\in(0,1)$. Hence in this interval, we have that $f$ is constant. Since $f(0)=f(1/2)=f(1)=\pi/2$, we conclude that $$\arcsin ( 1 - 2x) + 2 \arcsin\sqrt x=\frac\pi2.$$

Answer 3

Let $x=u^2$ with $0\le u\le1$ and rewrite the equation $\arcsin(1-2x)+2\arcsin(\sqrt x)=\pi/2$ as

$$\arcsin(1-2u^2)={\pi\over2}-2\arcsin u$$

Think of each side as an angle, $\theta_L$ and $\theta_R$. Clearly $-\pi/2\le\theta_L\lt\pi/2$, that being the range of the arcsine function. Because $u$ is nonnegative, we have $0\le2\arcsin u\le\pi$, and hence $-\pi/2\le\theta_R\le\pi/2$ as well. Thus to prove that $\theta_L=\theta_R$, it suffices to show that $\sin\theta_L=\sin\theta_R$. Obviously

$$\sin\theta_L=\sin(\arcsin(1-2u^2))=1-2u^2$$

while standard trig identities take care of $\theta_R$:

$$\sin\theta_R=\sin\left({\pi\over2}-2\arcsin u \right)=\cos(2\arcsin u)=1-2\sin^2(\arcsin u)=1-2u^2$$