Prove the identity:

\[ \cos(A + B + C) = \cos A \cos B \cos C - \cos A \sin B \sin C - \sin A \cos B \cos C - \sin A \sin B \cos C \]

Hello !
cos (a+b) = cos a cos b - sin a sin b sin (a+b) = sin a cos b + sin b cos a
cos (a+b+c) = cos (a+(b+c)) cos (a+b+c) = cos a cos (b+c) - sin a sin (b+c) cos (a+b+c) = cos a (cos b cos c - sin b sin c) - sin a (sin b cos c + sin c cos b) cos (a+b+c)=cos a cos b cos c - cos a sin b sin c - sin a sin b cos c - sin a cos b sin c

Answered by xxx102 | 2024-06-10

The identity cos ( A + B + C ) = cos A cos B cos C − cos A sin B sin C − sin A cos B cos C − sin A sin B cos C is proven by using the cosine addition formulas. We expand and combine terms step-by-step to arrive at the original expression. Therefore, the identity holds true under these transformations.
;

Answered by xxx102 | 2024-09-03