The value of \(\cos ^2 10^{\circ}-\cos 10^{\circ} \cdot \cos 50^{\circ}+\cos ^2 50^{\circ}\)

\( \cos ^2 10^{\circ}-\cos 10^{\circ} \cdot \cos 50^{\circ}+\cos ^2 50^{\circ}=\)\(\frac{\cos ^3 10^{\circ}+\cos ^3 50^{\circ}}{\cos 10^{\circ}+\cos ^{\circ}} \)
\( =\frac{\frac{3 \cos 10^{\circ}+\cos 30^{\circ}}{4}+\frac{3 \cos 50^{\circ}+\cos 150^{\circ}}{4}}{\cos 10^{\circ}+\cos 50^{\circ}} \)
\( {\left[\because \cos ^3 \mathrm{~A}=\frac{3 \cos \mathrm{A}+\cos 3 \mathrm{~A}}{4}\right]} \)
\( =\frac{\frac{3}{4}\left(\cos 10^{\circ}+\frac{\cos 30^{\circ}}{3}+\cos 50^{\circ}-\frac{\cos 30^{\circ}}{4}\right)}{\cos 10^{\circ}+\cos 50^{\circ}} \)
\( =\frac{3}{4} \frac{\left(\cos 10^{\circ}+\cos 50^{\circ}\right)}{\left(\cos 10^{\circ}+\cos 50^{\circ}\right)} \)
\( =\frac{3}{4} \)