If ${cos}^{2} 2 x - {cos}^{2} 6 x = sin m x sin 8 x$ . Find $m$

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Solution

$c o s^{2} 2 x - c o s^{2} 6 x = (c o s 2 x + c o s 6 x) (c o s 2 x - c o s 6 x) = 2 c o s ((\frac{2 x + 6 x}{2})) c o s ((\frac{2 x - 6 x}{2})) \cdot 2 s i n (\frac{2 x + 6 x}{2}) \cdot s i n ((\frac{2 x - 6 x}{2})) = 4 c o s 4 x c o s 2 x s i n 4 x s i n 2 x = (2 s i n 4 x c o s 4 x) (2 s i n 2 x c o s 2 x) = s i n 8 x s i n 4 x ∴ m = 4$

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