$\frac{cos 5 x + cos 4 x}{1 - 2 cos 3 x}$ $=$ $- (cos 2 x + cos x)$

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Solution

$L H S = \frac{(c o s 5 x + c o s 4 x)}{(1 - 2 c o s 3 x)}$
$= \frac{2 c o s (9 x / 2) c o s (x / 2)}{(1 - 2 c o s 3 x)}$
Above and below multiple by $c o s (3 x / 2)$
$= \frac{(2 c o s 9 x / 2 c o s x / 2 c o s 3 x / 2)}{(c o s 3 x / 2 - 2 c o s 3 x c o s 3 x / 2)}$
$= \frac{(2 c o s 9 x / 2 c o s x / 2 c o s 3 x / 2)}{(c o s 3 x / 2 - c o s 9 x / 2 - 2 c o s 3 x / 2)}$
$= \frac{(2 c o s 9 x / 2 c o s x / 2 c o s 3 x / 2)}{(- c o s 9 x / 2)}$
$= - 2 c o s (3 x / 2) c o s (x / 2)$
$= - (c o s 2 x + c o s x) R H S .$

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