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Expansion of cos (A + B + C)
We will learn how to find the expansion of cos (A + B + C). By using the formula of cos (α + β) and sin (α + β) we can easily expand cos (A + B + C).
Let us recall the formula of cos (α + β) = cos α cos β - sin α sin β and sin (α + β) = sin α cos β + cos α sin β.
cos (A + B + C) = cos [(A + B) + C]
= cos (A + B) cos C - sin (A + B) sin C, [applying the formula of cos (α + β)]
= (cos A cos B - sin A sin B) cos C - (sin A cos B + cos A sin B) sin C, [applying the formula of cos (α + β) and sin (α + β)]
= cos A cos B cos C - sin A sin B sin C - sin C sin A cos B - sin B sin C cos A, [applying distributive property]
= cos A cos B cos C (1 - tan A tan B - tan C tan A - tan B tan C)
Therefore, the expansion of cos (A + B + C) = cos A cos B cos C (1 - tan A tan B - tan C tan A - tan B tan C)
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