Solving for $x$ in $\sin^{-1}(2x) + \sin^{-1}(3x) = \frac \pi 4$

There is a useful identity that we can use in this case:

$\arcsin{x}+\arcsin{y}=\arcsin{(x\sqrt{1-y^2}+y\sqrt{1-x^2})}$

From here we can substitute:

$\arcsin{(2x\sqrt{1-9x^2}+3x\sqrt{1-4x^2})}=\frac{\pi}{4}$

We are then left with:

$2x\sqrt{1-9x^2}+3x\sqrt{1-4x^2}=\sin{\frac{\pi}{4}}$

From here, you can solve for $x$.

Let $\theta_1 = \sin^{-1}(2x)$,$\theta_2 = \sin^{-1}(3x)$. Then, $$\sin(\theta_1+\theta_2) = \sin(\theta_1)\cos(\theta_2)+\sin(\theta_2)\cos(\theta_1) = \sin(\pi/4) = 1/\sqrt{2}$$ $$2x\sqrt{1-9x^2}+3x\sqrt{1-4x^2} = \frac{1}{\sqrt{2}}.$$ But I don't know how you would solve this last equation.

I'm gonna derive the general function for $\arcsin x$ then go from there.

Recall that $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$ So if $y= \arcsin x$, then $$2ix=e^{iy}-e^{-iy}$$ Letting $u=e^{iy}$, we have $$2ix=\frac{u^2-1}{u}$$ $$u^2-2ixu-1=0$$ Use the quadratic formula to find that $$u=ix+\sqrt{1-x^2}$$ Thus $$e^{iy}=ix+\sqrt{1-x^2}$$ $$iy=\ln\big[ix+\sqrt{1-x^2}\big]$$ $$\arcsin x=-i\ln\big[ix+\sqrt{1-x^2}\big]$$ So we look at your equation: $$\arcsin 2x+\arcsin3x=\frac\pi4$$ $$-i\ln\big[2ix+\sqrt{1-4x^2}\big]-i\ln\big[3ix+\sqrt{1-9x^2}\big]=\frac\pi4$$ $$\ln\big[2ix+\sqrt{1-4x^2}\big]+\ln\big[3ix+\sqrt{1-9x^2}\big]=\frac{i\pi}4$$ Using the property $\ln(ab)=\ln a+\ln b$ we see that $$\ln\bigg[\big(2ix+\sqrt{1-4x^2}\big)\big(3ix+\sqrt{1-9x^2}\big)\bigg]=\frac{i\pi}4$$ Taking $\exp$ on both sides, $$\big(2ix+\sqrt{1-4x^2}\big)\big(3ix+\sqrt{1-9x^2}\big)=e^{i\pi/4}$$ Use the formula $e^{i\theta}=\cos\theta+i\sin\theta$ to see that $$\big(2ix+\sqrt{1-4x^2}\big)\big(3ix+\sqrt{1-9x^2}\big)=\frac{1+i}{\sqrt2}$$ and at this point I used Wolfram|Alpha to see that $$x=\sqrt{\frac{13}{194}-\frac{3\sqrt2}{97}}$$ I will update my answer once I figure out how this result is found

---

Edit:

Expanding out the product on the Left hand side, then multiplying both sides by $-i$, we have $$x\bigg(2\sqrt{1-9x^2}+3\sqrt{1-4x^2}\bigg)+6ix^2-i\sqrt{(1-4x^2)(1-9x^2)}=\frac{1-i}{\sqrt{2}}$$ We set the real parts of each side equal to eachother: $$x\bigg(2\sqrt{1-9x^2}+3\sqrt{1-4x^2}\bigg)=\frac1{\sqrt2}$$ Which @ClaudeLeibovici showed reduced to $$97y^2-13y+\frac14=0$$ with $y=x^2$. Using the quadratic formula, we see that $$y=\frac{13+\sqrt{72}}{194}$$ which reduces to $$y=\frac{13}{194}+\frac{3\sqrt2}{97}$$ Taking $\sqrt{\cdot}$ on both sides, $$x=\sqrt{\frac{13}{194}-\frac{3\sqrt2}{97}}$$