$5.43 \cos \left(22^{\circ}-t\right) \cos \left(82^{\circ}-t\right)+\cos \left(112^{\circ}-t\right) \cos \left(172^{\circ}-t\right)=$ $=0.5(\sin t+\cos t)$.
5.43 Transform the left side of the equation:

$$
\begin{aligned}
& \frac{1}{2} \cos \left(104^{\circ}-2 t\right)+\frac{1}{2} \cos 60^{\circ}+\frac{1}{2} \cos \left(284^{\circ}-2 t\right)+\frac{1}{2} \cos 60^{\circ}= \\
& =-\frac{1}{2} \sin \left(14^{\circ}-2 t\right)+\frac{1}{4}+\frac{1}{2} \sin \left(14^{\circ}-2 t\right)+\frac{1}{4}=\frac{1}{2}
\end{aligned}
$$

Then we get the equation

$$
\sin t+\cos t=1 ; \sin t=2 \sin ^{2} \frac{t}{2} ; 2 \sin \frac{t}{2} \cos \frac{t}{2}=2 \sin ^{2} \frac{t}{2}
$$

1) $\sin \frac{t}{2}=0 ; t_{1}=360^{\circ} \cdot k$;
2) $\sin \frac{t}{2}=\cos \frac{t}{2}, \operatorname{tg} \frac{t}{2}=1 ; t_{2}=90^{\circ}(4 k+1)$.

Answer: $t_{1}=360^{\circ} \cdot k, t_{2}=90^{\circ}(4 k+1), k \in Z$.