Question

Find the expansion of
cos ((A-B)-C)​

Answer 1

Answer:

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)