Question

Prove that $4\cos x\times \cos \left(\frac{\pi }{3}+x\right)\cos \left(\frac{\pi }{3}-x\right)=\cos 3x$

Answer 1

$\begin{array}{c}Wehavetoprove\\ 4\cos x\times \cos \left(\frac{\pi }{3}+x\right)\cos \left(\frac{\pi }{3}-x\right)=\cos 3x\\ Now,\\ L.H.S=4\cos x\times \cos \left(\frac{\pi }{3}+x\right)\cos \left(\frac{\pi }{3}-x\right)\\ =4\cos x\times \left(co{s}^2\frac{\pi }{3}-{\sin }^{2}x\right)\\ =4\cos x\times \left(\frac{1}{4}-{\sin }^{2}x\right)\\ =4\cos x\times \left(\frac{1}{4}-1+{\cos }^{2}x\right)\\ =4\cos x\times \left({\cos }^{2}x-\frac{3}{4}\right)\\ =\cos x\times \left(4{\cos }^{2}x-3\right)\\ ={\cos }^{3}x-3\cos x\\ =\cos 3x\end{array}$.