Question

Prove that: $4\left({\cos }^{3}10°+{\sin }^{3}20°\right)=3\left(\cos 10°+\sin 20°\right)$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know}, \\ \sin 60°=\cos 30°\left(=\frac{\sqrt{3}}{2}\right) \\ \Rightarrow \sin 3\times 20°=\cos 3\times 10° \\ \Rightarrow 3\sin 20°-4{\sin }^{3}20°=4{\cos }^{3}10°-3\cos 10° \\ \left(\because \sin 3\mathrm{\theta }=3\mathrm{sin\theta }-4{\sin }^3\mathrm{\theta }\mathrm{and}\cos 3\mathrm{\theta }=4{\cos }^3\mathrm{\theta }-3\mathrm{cos\theta }\right) \\ \Rightarrow 4\left({\cos }^{3}10+{\sin }^{3}20°\right)=3\left(\cos 10°+\sin 20°\right) \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$