Prove the following:
$c o s^{2} 2 x - c o s^{2} 6 x = s i n 4 x s i n 8 x$

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Solution

$L H S = c o s^{2} 2 x - c o s^{2} 6 x$
$= (cos 2 x - cos 6 x) (cos 2 x + cos 6 x)$
$= (- 2 sin (\frac{2 x + 6 x}{2}) sin (\frac{2 x - 6 x}{2})) (- 2 cos (\frac{2 x + 6 x}{2}) cos (\frac{2 x - 6 x}{2})) = (2 sin 4 x cos 4 x) (2 sin 2 x cos 2 x) = sin 4 x sin 8 x$

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