Problem 6. Solve the equation $\arcsin \frac{x \sqrt{11}}{2 \sqrt{21}}+\arcsin \frac{x \sqrt{11}}{4 \sqrt{21}}=\arcsin \frac{5 x \sqrt{11}}{8 \sqrt{21}}$.
Answer: $0, \pm \frac{21}{10}$

Solution. From the condition on the domain of the arcsine, it follows that

$$
|x| \leqslant \frac{8 \sqrt{21}}{5 \sqrt{11}}, \text { or, equivalently, } x^{2} \leqslant \frac{1344}{275} .
$$

By calculating the sine of both sides of the equation and considering that

$$
\cos \arcsin t>0
$$

and, consequently,

$$
\sin (\arcsin t)=t, \text { and } \cos (\arcsin t)=\sqrt{1-t^{2}},
$$

we obtain

$$
\frac{x \sqrt{11}}{2 \sqrt{21}} \cdot \sqrt{1-\frac{11 x^{2}}{16 \cdot 21}}+\frac{x \sqrt{11}}{4 \sqrt{21}} \cdot \sqrt{1-\frac{11 x^{2}}{4 \cdot 21}}=\frac{5 x \sqrt{11}}{8 \sqrt{21}}
$$

By moving everything to the left side of the equation, simplifying, and factoring out the common factor, we have

$$
\frac{x \sqrt{11}}{8 \cdot 21}\left(\sqrt{336-11 x^{2}}+\sqrt{84-11 x^{2}}-5 \sqrt{21}\right)=0
$$

From this equation, it follows that either $x=0$ (which obviously fits), or $x$ is a root of the equation

$$
\sqrt{336-11 x^{2}}+\sqrt{84-11 x^{2}}=5 \sqrt{21}
$$

From condition (*) it follows that all expressions under the square roots are positive. Since both sides of the equation are positive, they can be squared

$$
336-11 x^{2}+2 \sqrt{\left(336-11 x^{2}\right)\left(84-11 x^{2}\right)}+84-11 x^{2}=525 .
$$

By moving everything except the square root to the right side of the equation, we have

$$
2 \sqrt{\left(336-11 x^{2}\right)\left(84-11 x^{2}\right)}=22 x^{2}+105
$$

By squaring both sides of the equation again, we get

$$
4\left(336-11 x^{2}\right)\left(84-11 x^{2}\right)=484 x^{4}+4620 x^{2}+11025
$$

or

$$
23100 x^{2}=101871
$$

Thus, the equation has two more possible roots

$$
x= \pm \frac{21}{10}
$$

Verification. We verify that the left side of the equation lies in the interval $[-\pi / 2 ; \pi / 2]$ for the given values of the argument. For this, we calculate the cosine of the left side

$$
\begin{aligned}
\cos \left(\arcsin \frac{x \sqrt{11}}{2 \sqrt{21}}+\arcsin \frac{x \sqrt{11}}{4 \sqrt{21}}\right. & =\sqrt{\left(1-\frac{11 x^{2}}{4 \cdot 21}\right)\left(1-\frac{11 x^{2}}{16 \cdot 21}\right)}-\frac{11 x^{2}}{8 \cdot 21}= \\
& =\sqrt{\left(1-\frac{11 \cdot 21}{4 \cdot 100}\right)\left(1-\frac{11 \cdot 21}{16 \cdot 100}\right)}-\frac{11 \cdot 21}{8 \cdot 100}=\frac{13 \cdot 37}{800}-\frac{11 \cdot 21}{800}>0
\end{aligned}
$$

Since the cosine value is positive, and the left side lies in the interval $[-\pi ; \pi]$, it lies in the interval $[-\pi / 2 ; \pi / 2]$. Therefore, all the found numbers are solutions to the problem.