Solve the following system of equations:

$$
\operatorname{ctg} x+\operatorname{tg} y=2
$$

$$
\sin x \cdot \cos y=\frac{1}{4}
$$
Solve the following system of equations:

$$
\operatorname{ctg} x+\operatorname{tg} y=2
$$

$$
\sin x \cdot \cos y=\frac{1}{4}
$$

From (1),

$$
\cos x \cos y+\sin x \sin y=2 \sin x \cos y
$$

Substituting (2) into this,

$$
\cos x \cos y+\sin x \sin y=\frac{1}{2}
$$

or

$$
\cos (x-y)=\frac{1}{2}
$$

so

$$
x-y=60^{\circ} \text {. }
$$

From (2),

$$
2 \sin x \cos y=\frac{1}{2}
$$

From (3),

$$
\sin x \cos y-\cos x \sin y=\frac{1}{2} \sqrt{3}
$$

Subtracting this from (4), we get

$$
\sin x \cos y+\cos x \sin y=\frac{1}{2}(1-\sqrt{3})
$$

or

$$
\sin (x+y)=\frac{1}{2}(1-\sqrt{3})=-\frac{1}{2}(\sqrt{3}-1)
$$

from which

$$
x+y=360^{\circ}-21^{\circ} 28^{\prime} 15^{\prime \prime}=338^{\circ} 31^{\prime} 45^{\prime \prime}
$$

and

$$
x+y=180^{\circ}+21^{\circ} 28^{\prime} 15^{\prime \prime}=201^{\circ} 28^{\prime} 15^{\prime \prime}
$$

Connecting these values with (3), we get

$$
x_{1}=199^{\circ} 15^{\prime} 53^{\prime \prime}, y_{1}=139^{\circ} 15^{\prime} 53^{\prime \prime}, x_{2}=130^{\circ} 44^{\prime} 7^{\prime \prime}, y_{2}=70^{\circ} 44^{\prime} 7^{\prime \prime}
$$

The problem was also solved by: Détshy K., Freibauer E., Goldziher K., Spitzer Ö., Weisz J.