Prove that: 4 ( 3 +x ) ( 3 -x )= 3x - Vidyadip

Question

Prove that: $4\cos\text{x}\cos\Big(\frac{\pi}{3}+\text{x}\Big)\cos\Big({\frac{\pi}{3}-\text{x}}\Big)=\cos3\text{x}$

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Answer

We have, $\text{LHS}=4\cos\text{x}\cos\Big(\frac{\pi}{3}+\text{x}\Big)\cos\Big(\frac{\pi}{3}-\text{x}\Big)$ $=\ 2\cos\text{x}\Big[2\cos\Big(\frac{\pi}{3}+\text{x}\Big)\cos\Big(\frac{\pi}{3}-\text{x}\Big)\Big]$ $=\ 2\cos\text{x}\Big[2\cos\Big(\frac{\pi}{3}+\text{x}+\frac{\pi}{3}-\text{x}\Big)+\cos\Big(\frac{\pi}{3}+\text{x}-\frac{\pi}{3}+\text{x}\Big)\Big]$ $=\ 2\cos\text{x}\Big[\cos\frac{2\pi}{3}+\cos2\text{x}\Big]$ $=\ 2\cos\text{x}\Big[\cos\Big(\frac{\pi}{2}+\frac{\pi}{6}\Big)+\cos2\text{x}\Big]$ $=\ 2\cos\text{x}\Big[-\sin\frac{\pi}{6}+\cos2\text{x}\Big]$ $=\ 2\cos\text{x}\Big[-\frac{1}{2}+\cos2\text{x}\Big]$ $=\ -2\cos\text{x}\times\frac{1}{2}+2\cos\text{x}\cos2\text{x}$ $=\ -\cos\text{x}+[\cos(\text{x}+2\text{x})+\cos(2\text{x}-\text{x})]$ $=\ -\cos\text{x}+\cos3\text{x}+\cos\text{x}$ $=\ \cos3\text{x}$ $=\ \text{RHS}$ LHS = RHS Hence Proved.

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