Prove the following: ^22x- ^26x= 8x 4x - Vidyadip

Question

Prove the following:
$\cos^22\text{x}-\cos^26\text{x}=\sin8\text{x}\sin4\text{x}$

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Answer

It is known that $\cos\text{A}+\cos\text{B}=2\cos\Big(\frac{\text{A}+\text{B}}{2}\Big).\cos\Big(\frac{\text{A}-\text{B}}{2}\Big).$ $\cos\text{A}-\cos\text{B}=-2\sin\Big(\frac{\text{A}+\text{B}}{2}\Big).\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)$ $\therefore\text{L.H.S.}=\cos^22\text{x}-\cos^26\text{x}$$=(\cos2\text{x}+\cos6\text{x})(\cos2\text{x}-6\text{x})$
$=\Big[2\cos\Big(\frac{2\text{x}+6\text{x}}{2}\Big)\cos\Big(\frac{2\text{x}-6\text{x}}{2}\Big)\Big]\Big[-2\sin\Big(\frac{2\text{x}+6\text{x}}{2}\Big)\sin\Big(\frac{2\text{x}-6\text{x}}{2}\Big)\Big]$
$=[2\cos4\text{x}\cos(-2\text{x})][-2\sin4\text{x}\sin(-2\text{x})]$
$=[2\cos4\text{x}\cos2\text{x}][-2\sin4\text{x})(\sin-2\text{x})]$
$=(2\cos4\text{x}\cos4\text{x})(2\sin2\text{x}\cos2\text{x})$ $=\sin8\text{x}\sin4\text{x}$ $=\text{R.H.S.}$

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