Which is the smallest positive $x$ angle for which

$$
2^{\sin ^{2} x} \cdot 4^{\cos ^{2} x} \cdot 2^{\tan x}=8
$$
Using the properties of exponentiation, we obtain the following trigonometric equation:

$$
\sin ^{2} x+2 \cos ^{2} x+\operatorname{tg} x=3
$$

By substituting the identities $\cos ^{2} x=1-\sin ^{2} x, \operatorname{tg} x=\frac{\sin x}{\cos x}$, we get

$$
\sin ^{2} x+2-2 \sin ^{2} x+\frac{\sin x}{\cos x}=3
$$

from which

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

Squaring the equation and applying the square identity again, we obtain

$$
\sin ^{6} x \sin ^{4} x-1=0 ; \quad \text { that is } \quad y=\sin ^{2} x
$$

By substituting, we arrive at the following cubic equation:

$$
y^{3}+y^{2}-1=0
$$

Some of the solvers graphically solved equation (1): they plotted the functions $\operatorname{tg} x$ and $1+\sin ^{2} x$ separately and read off the approximate root from the graph.

Others determined the value of the angle using iteration (approximation method) on a calculator. Those who knew and could apply the Cardano formula were able to calculate the value of $x$, $x \approx 1.0530$ radians, or about $60^{\circ}$.

Those solvers who successfully determined the approximate value of the roots using any of the above methods received 2 points.

Remarks 1. The Cardano formula is a similar relationship to the quadratic formula. The general $a x^{3}+b x^{2}+c x+d=0$ cubic equation can be transformed into the form $y^{3}+p y+q=0$ by substitution, where $p$ and $q$ can be expressed using the four basic operations with the coefficients $a, b, c, d$.

The solutions of the equation are given by the

$$
y=\sqrt[3]{-\frac{q}{2}+\sqrt{\left(\frac{q}{2}\right)^{2}+\left(\frac{p}{3}\right)^{3}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\left(\frac{q}{2}\right)^{2}+\left(\frac{p}{3}\right)^{3}}}
$$

formula. This is the so-called Cardano formula. (For more information, see e.g. Great Moments in the History of Mathematics, Gondolat Publishing, Bp., 1981, pp. 58-66.)

2. The solution of the quadratic equation was already known to the Greeks, while the solution of cubic and quartic equations became known only in the 16th century. G. Cardano, an Italian mathematician, published the formula for solving the cubic equation in his book in 1539. L. Ferrari, also an Italian mathematician, was the first to provide a solution method for the quartic equation. Many mathematicians tried to solve higher-degree equations, that is, to find a formula that could solve any higher-degree equation. Finally, N. H. Abel, a Norwegian mathematician, proved that there is no such formula for the general fifth-degree and higher-degree equations.
3. A regrettable error crept into the problem. Originally, we intended to determine the smallest root of the equation

$$
2^{\sin ^{2} x} \cdot 4^{\cos ^{2} x} \cdot 2^{\operatorname{tg}^{2} x}=8
$$

This problem is significantly simpler, and we encourage our readers to try it.