Question

Prove that 4 cos 12 degrees × cos 48 degrees × cos 72 degrees = cos 36 degrees.

Answer 1

$$\begin{gathered} \mathrm{LHS}=4\cos 12°\cos 48°\cos 72° \\ =2\cos 12°\left[\cos \left(72°+48°\right)+\cos \left(72°-48°\right)\right]\mathbf{}\mathbf{}\mathbf{}\left[\mathbf{Since}\mathbf{}\mathbf{2}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{A}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{B}\mathbf{=}\mathbf{cos}\left(\mathbf{A}\mathbf{+}\mathbf{B}\right)\mathbf{+}\mathbf{cos}\left(\mathbf{A}\mathbf{-}\mathbf{B}\right)\right] \\ =2\cos 12°\left(\cos 120°+\cos 24°\right) \\ =2\cos 12°\left(-\frac{1}{2}+\cos 24°\right) \\ =-\cos 12°+2\cos 12°\cos 24° \\ =-\cos 12°+\left[\cos \left(24°+12°\right)+\cos \left(24°-12°\right)\right] \\ =-\cos 12°+\cos 36°+\cos 12° \\ =\cos 36°=\mathrm{RHS} \end{gathered}$$