Prove the following:
$s i n (n + 1) x s i n (n + 2) x + c o s (n + 1) x c o s (n + 2) x = c o s x$

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Solution

Use $c o s A c o s B + s i n A s i n B = c o s (A - B)$

$L H S = s i n (n + 1) x s i n (n + 2) x + c o s (n + 1) x c o s (n + 2) x = cos [(n + 2) x - (n + 1) x]$

$= cos x = R H S$

Hence proved.

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