3. Calculate the value of the expression $\sin ^{8} 75^{\circ}-\cos ^{8} 75^{\circ}$.
III/3. Let's calculate

$$
\begin{aligned}
\sin ^{8} 75^{\circ}-\cos ^{8} 75^{\circ}= & \left(\sin ^{4} 75^{\circ}-\cos ^{4} 75^{\circ}\right)\left(\sin ^{4} 75^{\circ}+\cos ^{4} 75^{\circ}\right)= \\
= & \left(\sin ^{2} 75^{\circ}-\cos ^{2} 75^{\circ}\right)\left(\sin ^{2} 75^{\circ}+\cos ^{2} 75^{\circ}\right) \\
& \cdot\left(\left(\sin ^{2} 75^{\circ}+\cos ^{2} 75^{\circ}\right)^{2}-2 \sin ^{2} 75^{\circ} \cos ^{2} 75^{\circ}\right)= \\
= & -\cos \left(2 \cdot 75^{\circ}\right)\left(1-\frac{1}{2} \sin ^{2}\left(2 \cdot 75^{\circ}\right)\right)=-\cos 150^{\circ}\left(1-\frac{1}{2} \sin ^{2} 150^{\circ}\right)= \\
= & \frac{\sqrt{3}}{2}\left(1-\frac{1}{8}\right)=\frac{7 \sqrt{3}}{16}
\end{aligned}
$$

Factorization into the difference of fourth powers: 2 points. Further factorization: 1 point. Simplification using the identity $\sin ^{2} 75^{\circ}+\cos ^{2} 75^{\circ}=1: \mathbf{1}$ point. Calculation: $\mathbf{3}$ points.

If the contestant does not solve the problem correctly but significantly simplifies the expression: up to 4 points. The use of a calculator is, of course, not allowed.