The value of ${cos}^{2} A + {cos}^{2} B - 2 cos A cos B cos (A + B) - {sin}^{2} (A + B)$

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Solution

L.H.S. = $c o s^{2} A + c o s^{2} B - 2 c o s A c o s B (c o s A c o s B - s i n A s i n B)$
$= (c o s^{2} A - c o s^{2} A c o s^{2} B) + (c o s^{2} B - c o s^{2} A c o s^{2} B) + 2 s i n A s i n B c o s A c o s B$
$= c o s^{2} A (1 - c o s^{2} B) + c o s^{2} B (1 - c o s^{2} A) + 2 s i n A s i n B c o s A c o s B$
$= c o s^{2} A s i n^{2} B + c o s^{2} B s i n^{2} A + 2 s i n A s i n B c o s A c o s B$
$= (s i n A c o s B + c o s A s i n B)^{2} = s i n^{2} (A + B)$

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