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

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Aicharya · Accepted Answer

$\begin{aligned} & \cos ^2 10^{\circ}-\cos 10^{\circ} \cdot \cos 50^{\circ}+\cos ^2 50^{\circ}=\frac{\cos ^3 10^{\circ}+\cos ^3 50^{\circ}}{\cos 10^{\circ}+\cos ^{\circ}} \ & =\frac{\frac{3 \cos 10^{\circ}+\cos 30^{\circ}}{4}+\frac{3 \cos 50^{\circ}+\cos 150^{\circ}}{4}}{\cos 10^{\circ}+\cos 50^{\circ}} \ & {\left[\because \cos ^3 \mathrm{~A}=\frac{3 \cos \mathrm{A}+\cos 3 \mathrm{~A}}{4}ight]} \ & =\frac{\frac{3}{4}\left(\cos 10^{\circ}+\frac{\cos 30^{\circ}}{3}+\cos 50^{\circ}-\frac{\cos 30^{\circ}}{4}ight)}{\cos 10^{\circ}+\cos 50^{\circ}} \ & =\frac{3}{4} \frac{\left(\cos 10^{\circ}+\cos 50^{\circ}ight)}{\left(\cos 10^{\circ}+\cos 50^{\circ}ight)} \ & =\frac{3}{4} \ & \end{aligned}$

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

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