22.3. (SRP, 66). Find all polynomials $R(x)$ of degree less than 4, for each of which there exists a polynomial $P(x)$ satisfying the identity $7 \sin ^{31} t+8 \sin ^{18} t-5 \sin ^{5} t \cos ^{4} t-10 \sin ^{2} t+5 \sin ^{5} t-2=$ $=P(\sin t)\left(\sin ^{4} t-(1+\sin t)\left(\cos ^{2} t-2\right)\right)+R(\sin t), \quad t \in \mathbb{R}$.
22.3. Let's make the substitutions $\cos ^{2} t=1-\sin ^{2} t$ and $\sin t=x$. We will obtain that the desired polynomial $R(x)$ of degree less than 4 is the remainder from the division of the polynomial $S(x)=7 x^{31}+8 x^{13}-5 x^{9}-2$ by the polynomial $Q(x)=x^{4}+x^{3}+x^{2}+x+1$, since the identity $S(x)=P(x) Q(x)+R(x)$ must be satisfied for all $x \in[-1 ; 1]$, and therefore for all $x \in \mathbb{C}$ (see Theorem 52). Since $(x-1) Q(x)=x^{5}-1$, there exist 4 distinct values of the variable $x$ that satisfy the condition $Q(x)=0$. For each of these values, we have the equalities $x^{5}=1$ and

$$
\begin{aligned}
R(x)=S(x)=7 x^{31}+ & 8 x^{13}-5 x^{9}-2= \\
& =7 x+8 x^{3}-5 x^{4}-2+5 Q(x)=13 x^{3}+5 x^{2}+12 x+3
\end{aligned}
$$

Thus, the polynomials $R(x)$ and $13 x^{3}+5 x^{2}+12 x+3$, whose degrees do not exceed 3, take the same values at four distinct points, and therefore they coincide.